<?xml version="1.0" encoding="US-ASCII"?>
<!-- Generated by DFIN XBRL Schema Document - http://www.dfinsolutions.com/ - on 26-08-2026 11:57:28 AM -->
<!-- Copyright(c) 2026 Donnelley Financial Solutions, Inc. All Rights Reserved. -->
<schema targetNamespace="http://www.bny.com/20260826" xmlns="http://www.w3.org/2001/XMLSchema" xmlns:bmifiii="http://www.bny.com/20260826" xmlns:dei="http://xbrl.sec.gov/dei/2025" xmlns:srt-types="http://fasb.org/srt-types/2025" xmlns:deprecated="http://www.xbrl.org/2009/role/deprecated" xmlns:dtr="http://www.xbrl.org/2009/dtr" xmlns:oef="http://xbrl.sec.gov/oef/2025" xmlns:ref="http://www.xbrl.org/2006/ref" xmlns:xbrldt="http://xbrl.org/2005/xbrldt" xmlns:xl="http://www.xbrl.org/2003/XLink" xmlns:us-gaap="http://fasb.org/us-gaap/2025" xmlns:oef-pre="http://xbrl.sec.gov/oef-pre/2025" xmlns:oef-lab="http://xbrl.sec.gov/oef-lab/2025" xmlns:link="http://www.xbrl.org/2003/linkbase" xmlns:country="http://xbrl.sec.gov/country/2025" xmlns:us-roles="http://fasb.org/us-roles/2025" xmlns:enum2="http://xbrl.org/2020/extensible-enumerations-2.0" xmlns:us-types="http://fasb.org/us-types/2025" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:dtr-types="http://www.xbrl.org/dtr/type/2024-01-31" xmlns:oef-def="http://xbrl.sec.gov/oef-def/2025" xmlns:negated="http://www.xbrl.org/2009/role/negated" xmlns:xbrldi="http://xbrl.org/2006/xbrldi" xmlns:srt="http://fasb.org/srt/2025" xmlns:oef-sr="http://xbrl.sec.gov/oef-sr/2025" xmlns:srt-roles="http://fasb.org/srt-roles/2025" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xbrli="http://www.xbrl.org/2003/instance" xmlns:stpr="http://xbrl.sec.gov/stpr/2025" elementFormDefault="qualified">
	<import namespace="http://www.xbrl.org/2003/instance" schemaLocation="http://www.xbrl.org/2003/xbrl-instance-2003-12-31.xsd"/>
	<import namespace="http://www.xbrl.org/2003/linkbase" schemaLocation="http://www.xbrl.org/2003/xbrl-linkbase-2003-12-31.xsd"/>
	<import namespace="http://xbrl.sec.gov/dei/2025" schemaLocation="https://xbrl.sec.gov/dei/2025/dei-2025.xsd"/>
	<import namespace="http://fasb.org/us-types/2025" schemaLocation="https://xbrl.fasb.org/us-gaap/2025/elts/us-types-2025.xsd"/>
	<import namespace="http://www.xbrl.org/dtr/type/2024-01-31" schemaLocation="https://www.xbrl.org/dtr/type/2024-01-31/types.xsd"/>   
	<import namespace="http://xbrl.sec.gov/country/2025" schemaLocation="https://xbrl.sec.gov/country/2025/country-2025.xsd"/>
	<import namespace="http://fasb.org/srt/2025" schemaLocation="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd"/>
	<import namespace="http://fasb.org/srt-types/2025" schemaLocation="https://xbrl.fasb.org/srt/2025/elts/srt-types-2025.xsd"/>
	<import namespace="http://xbrl.sec.gov/oef-sr/2025" schemaLocation="https://xbrl.sec.gov/oef/2025/oef-sr-2025.xsd"/>
	<import namespace="http://fasb.org/us-gaap/2025" schemaLocation="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd"/>
	<import namespace="http://fasb.org/us-roles/2025" schemaLocation="https://xbrl.fasb.org/us-gaap/2025/elts/us-roles-2025.xsd"/>
	<import namespace="http://fasb.org/srt-roles/2025" schemaLocation="https://xbrl.fasb.org/srt/2025/elts/srt-roles-2025.xsd"/>
	<annotation>
		<appinfo>
			<link:linkbase id="lnk">
				<link:arcroleRef arcroleURI="http://xbrl.org/int/dim/arcrole/domain-member" xlink:type="simple" xlink:href="http://www.xbrl.org/2005/xbrldt-2005.xsd#domain-member"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/ShareholderReport" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#ShareholderReport"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/ShareholderReportHoldingsDetails" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#ShareholderReportHoldingsDetails"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/ShareholderReportLineGraphDetails" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#ShareholderReportLineGraphDetails"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/ShareholderReportAverageAnnualReturnDetails" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#ShareholderReportAverageAnnualReturnDetails"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/ClassOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#ClassOnly"/>
				<link:roleRef roleURI="http://fasb.org/us-gaap/role/eedm/ExtensibleEnumerationLists" xlink:type="simple" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-roles-2025.xsd#eedm"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/LargestHoldingOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#LargestHoldingOnly"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/Other1CategoryOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#Other1CategoryOnly"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/AdditionalIndexOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#AdditionalIndexOnly"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/BroadBasedIndexOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#BroadBasedIndexOnly"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/Other2CategoryOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#Other2CategoryOnly"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/SalesLoadOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#SalesLoadOnly"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/MaturityOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#MaturityOnly"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/IndustrySectorOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#IndustrySectorOnly"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/CreditQualityOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#CreditQualityOnly"/>
				<link:roleRef roleURI="http://xbrl.sec.gov/oef/role/GeographicRegionOnly" xlink:type="simple" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#GeographicRegionOnly"/>
				<link:presentationLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/ShareholderReport">              
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_ShareClassDomain" xlink:label="ShareClassDomain"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000793Member" xlink:label="C000000793Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000793Member" order="2.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000792Member" xlink:label="C000000792Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000792Member" order="3.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000790Member" xlink:label="C000000790Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000790Member" order="4.0" />
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				<link:presentationLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/ShareholderReportHoldingsDetails">               
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_ShareClassDomain" xlink:label="ShareClassDomain"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000793Member" xlink:label="C000000793Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000793Member" order="2.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000792Member" xlink:label="C000000792Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000792Member" order="3.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000790Member" xlink:label="C000000790Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000790Member" order="4.0" />   
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentTypeCategorizationMember" xlink:label="InvestmentTypeCategorizationMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_UnfundedFloatingRateLoanInterestsCommitmentsMember" xlink:label="UnfundedFloatingRateLoanInterestsCommitmentsMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="UnfundedFloatingRateLoanInterestsCommitmentsMember" order="2.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ForwardForeignCurrencyExchangeContractsMember" xlink:label="ForwardForeignCurrencyExchangeContractsMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="ForwardForeignCurrencyExchangeContractsMember" order="3.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_EquitySecuritiesPreferredStocksMember" xlink:label="EquitySecuritiesPreferredStocksMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="EquitySecuritiesPreferredStocksMember" order="4.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_EquitySecuritiesCommonStocksMember" xlink:label="EquitySecuritiesCommonStocksMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="EquitySecuritiesCommonStocksMember" order="5.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_FixedRateLoanInterestsMember" xlink:label="FixedRateLoanInterestsMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="FixedRateLoanInterestsMember" order="6.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ConvertibleCorporateBondsAndNotesMember" xlink:label="ConvertibleCorporateBondsAndNotesMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="ConvertibleCorporateBondsAndNotesMember" order="7.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_FloatingRateLoanInterestsMember" xlink:label="FloatingRateLoanInterestsMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="FloatingRateLoanInterestsMember" order="8.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_CorporateBondsAndNotesMember" xlink:label="CorporateBondsAndNotesMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="CorporateBondsAndNotesMember" order="9.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_NetOtherAssetsAndLiabilitiesMember" xlink:label="NetOtherAssetsAndLiabilitiesMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="NetOtherAssetsAndLiabilitiesMember" order="10.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_InvestmentCompaniesMember" xlink:label="InvestmentCompaniesMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="InvestmentCompaniesMember" order="11.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AssetBackedSecuritiesMember" xlink:label="AssetBackedSecuritiesMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="AssetBackedSecuritiesMember" order="12.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CollateralizedLoanObligationsMember" xlink:label="CollateralizedLoanObligationsMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="InvestmentTypeCategorizationMember" xlink:to="CollateralizedLoanObligationsMember" order="13.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_MaturityDomain" xlink:label="MaturityDomain"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_OtherCategory1Domain" xlink:label="OtherCategory1Domain"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_OtherCategory2Domain" xlink:label="OtherCategory2Domain"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_LargestHoldingsDomain" xlink:label="LargestHoldingsDomain"/>
					<!--GeographicRegionDomain Only presentation and label linkbases-->			 
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_GeographicRegionDomain" xlink:label="GeographicRegionDomain"/>
					<!--IndustrySectorDomain Only presentation and label linkbases-->	
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_IndustrySectorDomain" xlink:label="IndustrySectorDomain"/>	
					<link:loc xlink:type="locator" xlink:href="#bmifiii_InvestmentCompaniesSectorMember" xlink:label="InvestmentCompaniesSectorMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="InvestmentCompaniesSectorMember" order="2.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_NetOtherAssetsAndLiabilitiesSectorMember" xlink:label="NetOtherAssetsAndLiabilitiesSectorMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="NetOtherAssetsAndLiabilitiesSectorMember" order="3.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ConsumerCyclicalMember" xlink:label="ConsumerCyclicalMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="ConsumerCyclicalMember" order="4.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ConsumerNonCyclicalMember" xlink:label="ConsumerNonCyclicalMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="ConsumerNonCyclicalMember" order="5.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_BasicMaterialsMember" xlink:label="BasicMaterialsMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="BasicMaterialsMember" order="6.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_UtilitiesSectorMember" xlink:label="UtilitiesSectorMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="UtilitiesSectorMember" order="7.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_IndustrialSectorMember" xlink:label="IndustrialSectorMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="IndustrialSectorMember" order="8.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_CommunicationsSectorMember" xlink:label="CommunicationsSectorMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="CommunicationsSectorMember" order="9.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TechnologySectorMember" xlink:label="TechnologySectorMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="TechnologySectorMember" order="10.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancialServicesSectorMember" xlink:label="FinancialServicesSectorMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="FinancialServicesSectorMember" order="11.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EnergySectorMember" xlink:label="EnergySectorMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="IndustrySectorDomain" xlink:to="EnergySectorMember" order="12.0" />
					<!--CreditQualityDomain Only presentation and label linkbases-->	
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_CreditQualityDomain" xlink:label="CreditQualityDomain"/>	
					<!--ExternalCreditRatingFitchMember Only presentation and label linkbases-->	
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_ExternalCreditRatingFitchMember" xlink:label="ExternalCreditRatingFitchMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="CreditQualityDomain" xlink:to="ExternalCreditRatingFitchMember" xbrldt:usable="false" order="1.0"/>
					<!--ExternalCreditRatingMoodysMember Only presentation and label linkbases-->	
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_ExternalCreditRatingMoodysMember" xlink:label="ExternalCreditRatingMoodysMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="CreditQualityDomain" xlink:to="ExternalCreditRatingMoodysMember" xbrldt:usable="false" order="2.1"/>
					<!--ExternalCreditRatingStandardPoorsMember Only presentation and label linkbases-->	
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_ExternalCreditRatingStandardPoorsMember" xlink:label="ExternalCreditRatingStandardPoorsMember"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="CreditQualityDomain" xlink:to="ExternalCreditRatingStandardPoorsMember" xbrldt:usable="false" order="3.1"/>
					<!--CreditRatingAMBestDomain Only presentation and label linkbases-->	
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_CreditRatingAMBestDomain" xlink:label="CreditRatingAMBestDomain"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="CreditQualityDomain" xlink:to="CreditRatingAMBestDomain" xbrldt:usable="false" order="4.1"/>
				</link:presentationLink>
				<link:presentationLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/ShareholderReportLineGraphDetails">            
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_ShareClassDomain" xlink:label="ShareClassDomain"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000793Member" xlink:label="C000000793Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000793Member" order="2.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000792Member" xlink:label="C000000792Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000792Member" order="3.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000790Member" xlink:label="C000000790Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000790Member" order="4.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_AdditionalIndexDomain" xlink:label="AdditionalIndexDomain"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_BroadBasedIndexDomain" xlink:label="BroadBasedIndexDomain"/>
				</link:presentationLink>
				<link:presentationLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/ShareholderReportAverageAnnualReturnDetails">            
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_ShareClassDomain" xlink:label="ShareClassDomain"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000793Member" xlink:label="C000000793Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000793Member" order="2.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000792Member" xlink:label="C000000792Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000792Member" order="3.0" />
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000790Member" xlink:label="C000000790Member"/>
					<link:presentationArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/parent-child" xlink:from="ShareClassDomain" xlink:to="C000000790Member" order="4.0" />
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_AdditionalIndexDomain" xlink:label="AdditionalIndexDomain"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_BroadBasedIndexDomain" xlink:label="BroadBasedIndexDomain"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_WithSalesLoadMember" xlink:label="WithSalesLoadMember"/>
				</link:presentationLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/ClassOnly">               
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_ShareClassDomain" xlink:label="ShareClassDomain"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000793Member" xlink:label="C000000793Member"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="ShareClassDomain" xlink:to="C000000793Member" order="1.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000792Member" xlink:label="C000000792Member"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="ShareClassDomain" xlink:to="C000000792Member" order="2.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000790Member" xlink:label="C000000790Member"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="ShareClassDomain" xlink:to="C000000790Member" order="3.0"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/Other1CategoryOnly">               
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_OtherCategory1Domain" xlink:label="OtherCategory1Domain"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/Other2CategoryOnly">               
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_OtherCategory2Domain" xlink:label="OtherCategory2Domain"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/LargestHoldingOnly">               
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_LargestHoldingsDomain" xlink:label="LargestHoldingsDomain"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/SalesLoadOnly">
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_WithSalesLoadMember" xlink:label="WithSalesLoadMember"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://fasb.org/us-gaap/role/eedm/ExtensibleEnumerationLists">               
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentTypeCategorizationMember" xlink:label="InvestmentTypeCategorizationMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_UnfundedFloatingRateLoanInterestsCommitmentsMember" xlink:label="UnfundedFloatingRateLoanInterestsCommitmentsMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="UnfundedFloatingRateLoanInterestsCommitmentsMember" order="1.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ForwardForeignCurrencyExchangeContractsMember" xlink:label="ForwardForeignCurrencyExchangeContractsMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="ForwardForeignCurrencyExchangeContractsMember" order="2.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_EquitySecuritiesPreferredStocksMember" xlink:label="EquitySecuritiesPreferredStocksMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="EquitySecuritiesPreferredStocksMember" order="3.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_EquitySecuritiesCommonStocksMember" xlink:label="EquitySecuritiesCommonStocksMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="EquitySecuritiesCommonStocksMember" order="4.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_FixedRateLoanInterestsMember" xlink:label="FixedRateLoanInterestsMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="FixedRateLoanInterestsMember" order="5.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ConvertibleCorporateBondsAndNotesMember" xlink:label="ConvertibleCorporateBondsAndNotesMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="ConvertibleCorporateBondsAndNotesMember" order="6.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_FloatingRateLoanInterestsMember" xlink:label="FloatingRateLoanInterestsMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="FloatingRateLoanInterestsMember" order="7.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_CorporateBondsAndNotesMember" xlink:label="CorporateBondsAndNotesMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="CorporateBondsAndNotesMember" order="8.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_NetOtherAssetsAndLiabilitiesMember" xlink:label="NetOtherAssetsAndLiabilitiesMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="NetOtherAssetsAndLiabilitiesMember" order="9.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_InvestmentCompaniesMember" xlink:label="InvestmentCompaniesMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="InvestmentCompaniesMember" order="10.0"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CollateralizedLoanObligationsMember" xlink:label="CollateralizedLoanObligationsMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="InvestmentTypeCategorizationMember" xlink:to="CollateralizedLoanObligationsMember" order="11.0"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/BroadBasedIndexOnly">               
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_BroadBasedIndexDomain" xlink:label="BroadBasedIndexDomain"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/AdditionalIndexOnly">              
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_AdditionalIndexDomain" xlink:label="AdditionalIndexDomain"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/MaturityOnly">
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_MaturityDomain" xlink:label="MaturityDomain"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/IndustrySectorOnly">
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_IndustrySectorDomain" xlink:label="IndustrySectorDomain"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_InvestmentCompaniesSectorMember" xlink:label="InvestmentCompaniesSectorMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="IndustrySectorDomain" xlink:to="InvestmentCompaniesSectorMember" order="1.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_NetOtherAssetsAndLiabilitiesSectorMember" xlink:label="NetOtherAssetsAndLiabilitiesSectorMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="IndustrySectorDomain" xlink:to="NetOtherAssetsAndLiabilitiesSectorMember" order="2.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ConsumerCyclicalMember" xlink:label="ConsumerCyclicalMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="IndustrySectorDomain" xlink:to="ConsumerCyclicalMember" order="3.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ConsumerNonCyclicalMember" xlink:label="ConsumerNonCyclicalMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="IndustrySectorDomain" xlink:to="ConsumerNonCyclicalMember" order="4.0"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_BasicMaterialsMember" xlink:label="BasicMaterialsMember"/>
					<link:definitionArc xlink:type="arc" xlink:arcrole="http://xbrl.org/int/dim/arcrole/domain-member" xlink:from="IndustrySectorDomain" xlink:to="BasicMaterialsMember" order="5.0"/>
				</link:definitionLink>
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/CreditQualityOnly">
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_CreditQualityDomain" xlink:label="CreditQualityDomain"/>
				</link:definitionLink>
				<!--GeographicRegionOnly use only for STPR-->
				<link:definitionLink xlink:type="extended" xlink:role="http://xbrl.sec.gov/oef/role/GeographicRegionOnly">
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/oef/2025/oef-2025.xsd#oef_GeographicRegionDomain" xlink:label="GeographicRegionDomain"/>
				</link:definitionLink>
				<link:labelLink xlink:type="extended" xlink:role="http://www.xbrl.org/2003/role/link">               
					<link:loc xlink:type="locator" xlink:href="#bmifiii_InvestmentCompaniesSectorMember" xlink:label="InvestmentCompaniesSectorMember"/>
					<link:label xlink:type="resource" xlink:label="label_InvestmentCompaniesSectorMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Investment Companies Sector [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="InvestmentCompaniesSectorMember" xlink:to="label_InvestmentCompaniesSectorMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_NetOtherAssetsAndLiabilitiesSectorMember" xlink:label="NetOtherAssetsAndLiabilitiesSectorMember"/>
					<link:label xlink:type="resource" xlink:label="label_NetOtherAssetsAndLiabilitiesSectorMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Net Other Assets and Liabilities Sector [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="NetOtherAssetsAndLiabilitiesSectorMember" xlink:to="label_NetOtherAssetsAndLiabilitiesSectorMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ConsumerCyclicalMember" xlink:label="ConsumerCyclicalMember"/>
					<link:label xlink:type="resource" xlink:label="label_ConsumerCyclicalMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Consumer Cyclical [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="ConsumerCyclicalMember" xlink:to="label_ConsumerCyclicalMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ConsumerNonCyclicalMember" xlink:label="ConsumerNonCyclicalMember"/>
					<link:label xlink:type="resource" xlink:label="label_ConsumerNonCyclicalMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Consumer Non cyclical [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="ConsumerNonCyclicalMember" xlink:to="label_ConsumerNonCyclicalMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_BasicMaterialsMember" xlink:label="BasicMaterialsMember"/>
					<link:label xlink:type="resource" xlink:label="label_BasicMaterialsMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Basic Materials [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="BasicMaterialsMember" xlink:to="label_BasicMaterialsMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_UnfundedFloatingRateLoanInterestsCommitmentsMember" xlink:label="UnfundedFloatingRateLoanInterestsCommitmentsMember"/>
					<link:label xlink:type="resource" xlink:label="label_UnfundedFloatingRateLoanInterestsCommitmentsMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Unfunded Floating Rate Loan Interests Commitments [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="UnfundedFloatingRateLoanInterestsCommitmentsMember" xlink:to="label_UnfundedFloatingRateLoanInterestsCommitmentsMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ForwardForeignCurrencyExchangeContractsMember" xlink:label="ForwardForeignCurrencyExchangeContractsMember"/>
					<link:label xlink:type="resource" xlink:label="label_ForwardForeignCurrencyExchangeContractsMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Forward Foreign Currency Exchange Contracts [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="ForwardForeignCurrencyExchangeContractsMember" xlink:to="label_ForwardForeignCurrencyExchangeContractsMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_EquitySecuritiesPreferredStocksMember" xlink:label="EquitySecuritiesPreferredStocksMember"/>
					<link:label xlink:type="resource" xlink:label="label_EquitySecuritiesPreferredStocksMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Equity Securities Preferred Stocks [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="EquitySecuritiesPreferredStocksMember" xlink:to="label_EquitySecuritiesPreferredStocksMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_EquitySecuritiesCommonStocksMember" xlink:label="EquitySecuritiesCommonStocksMember"/>
					<link:label xlink:type="resource" xlink:label="label_EquitySecuritiesCommonStocksMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Equity Securities Common Stocks [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="EquitySecuritiesCommonStocksMember" xlink:to="label_EquitySecuritiesCommonStocksMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_FixedRateLoanInterestsMember" xlink:label="FixedRateLoanInterestsMember"/>
					<link:label xlink:type="resource" xlink:label="label_FixedRateLoanInterestsMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Fixed Rate Loan Interests [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="FixedRateLoanInterestsMember" xlink:to="label_FixedRateLoanInterestsMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_ConvertibleCorporateBondsAndNotesMember" xlink:label="ConvertibleCorporateBondsAndNotesMember"/>
					<link:label xlink:type="resource" xlink:label="label_ConvertibleCorporateBondsAndNotesMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Convertible Corporate Bonds and Notes [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="ConvertibleCorporateBondsAndNotesMember" xlink:to="label_ConvertibleCorporateBondsAndNotesMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_FloatingRateLoanInterestsMember" xlink:label="FloatingRateLoanInterestsMember"/>
					<link:label xlink:type="resource" xlink:label="label_FloatingRateLoanInterestsMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Floating Rate Loan Interests [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="FloatingRateLoanInterestsMember" xlink:to="label_FloatingRateLoanInterestsMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_CorporateBondsAndNotesMember" xlink:label="CorporateBondsAndNotesMember"/>
					<link:label xlink:type="resource" xlink:label="label_CorporateBondsAndNotesMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Corporate Bonds and Notes [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="CorporateBondsAndNotesMember" xlink:to="label_CorporateBondsAndNotesMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_NetOtherAssetsAndLiabilitiesMember" xlink:label="NetOtherAssetsAndLiabilitiesMember"/>
					<link:label xlink:type="resource" xlink:label="label_NetOtherAssetsAndLiabilitiesMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Net Other Assets and Liabilities [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="NetOtherAssetsAndLiabilitiesMember" xlink:to="label_NetOtherAssetsAndLiabilitiesMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_InvestmentCompaniesMember" xlink:label="InvestmentCompaniesMember"/>
					<link:label xlink:type="resource" xlink:label="label_InvestmentCompaniesMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Investment Companies [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="InvestmentCompaniesMember" xlink:to="label_InvestmentCompaniesMember"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000793Member" xlink:label="C000000793Member"/>
					<link:label xlink:type="resource" xlink:label="label_C000000793Member" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">C000000793 [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="C000000793Member" xlink:to="label_C000000793Member"/>
										<link:label xlink:type="resource" xlink:label="label_C000000793Member_2" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xml:lang="en-US">Class I</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="C000000793Member" xlink:to="label_C000000793Member_2"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000792Member" xlink:label="C000000792Member"/>
					<link:label xlink:type="resource" xlink:label="label_C000000792Member" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">C000000792 [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="C000000792Member" xlink:to="label_C000000792Member"/>
										<link:label xlink:type="resource" xlink:label="label_C000000792Member_2" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xml:lang="en-US">Class C</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="C000000792Member" xlink:to="label_C000000792Member_2"/>
					<link:loc xlink:type="locator" xlink:href="#bmifiii_C000000790Member" xlink:label="C000000790Member"/>
					<link:label xlink:type="resource" xlink:label="label_C000000790Member" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">C000000790 [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="C000000790Member" xlink:to="label_C000000790Member"/>
										<link:label xlink:type="resource" xlink:label="label_C000000790Member_2" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xml:lang="en-US">Class A</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="C000000790Member" xlink:to="label_C000000790Member_2"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AssetBackedSecuritiesMember" xlink:label="AssetBackedSecuritiesMember"/>
					<link:label xlink:type="resource" xlink:label="label_AssetBackedSecuritiesMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Asset Backed Securities [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="AssetBackedSecuritiesMember" xlink:to="label_AssetBackedSecuritiesMember"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TechnologySectorMember" xlink:label="TechnologySectorMember"/>
					<link:label xlink:type="resource" xlink:label="label_TechnologySectorMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Technology Sector [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="TechnologySectorMember" xlink:to="label_TechnologySectorMember"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancialServicesSectorMember" xlink:label="FinancialServicesSectorMember"/>
					<link:label xlink:type="resource" xlink:label="label_FinancialServicesSectorMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Financial Services Sector [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="FinancialServicesSectorMember" xlink:to="label_FinancialServicesSectorMember"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EnergySectorMember" xlink:label="EnergySectorMember"/>
					<link:label xlink:type="resource" xlink:label="label_EnergySectorMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Energy Sector [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="EnergySectorMember" xlink:to="label_EnergySectorMember"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CollateralizedLoanObligationsMember" xlink:label="CollateralizedLoanObligationsMember"/>
					<link:label xlink:type="resource" xlink:label="label_CollateralizedLoanObligationsMember" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Collateralized Loan Obligations [Member]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="CollateralizedLoanObligationsMember" xlink:to="label_CollateralizedLoanObligationsMember"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentTypeAxis" xlink:label="InvestmentTypeAxis"/>
					<link:label xlink:type="resource" xlink:label="label_InvestmentTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Investment Type [Axis]</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="InvestmentTypeAxis" xlink:to="label_InvestmentTypeAxis" priority="1"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AssetsNet" xlink:label="AssetsNet"/>
					<link:label xlink:type="resource" xlink:label="label_AssetsNet" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Net Assets</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="AssetsNet" xlink:to="label_AssetsNet" priority="1"/>
					<link:loc xlink:type="locator" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentCompanyPortfolioTurnover" xlink:label="InvestmentCompanyPortfolioTurnover"/>
					<link:label xlink:type="resource" xlink:label="label_InvestmentCompanyPortfolioTurnover" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">Investment Company Portfolio Turnover</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="InvestmentCompanyPortfolioTurnover" xlink:to="label_InvestmentCompanyPortfolioTurnover" priority="1"/>                
					<link:loc xlink:type="locator" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_NoTradingSymbolFlag" xlink:label="dei_NoTradingSymbolFlag"/>
					<link:label xlink:type="resource" xlink:label="label_NoTradingSymbolFlag" xlink:role="http://www.xbrl.org/2003/role/label" xml:lang="en-US">No Trading Symbol Flag</link:label>
					<link:labelArc xlink:type="arc" xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="dei_NoTradingSymbolFlag" xlink:to="label_NoTradingSymbolFlag" priority="1"/>		
				</link:labelLink>
			</link:linkbase>
		</appinfo>
	</annotation>   
	<element name="InvestmentCompaniesSectorMember" id="bmifiii_InvestmentCompaniesSectorMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="NetOtherAssetsAndLiabilitiesSectorMember" id="bmifiii_NetOtherAssetsAndLiabilitiesSectorMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="ConsumerCyclicalMember" id="bmifiii_ConsumerCyclicalMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="ConsumerNonCyclicalMember" id="bmifiii_ConsumerNonCyclicalMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="BasicMaterialsMember" id="bmifiii_BasicMaterialsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="UnfundedFloatingRateLoanInterestsCommitmentsMember" id="bmifiii_UnfundedFloatingRateLoanInterestsCommitmentsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="ForwardForeignCurrencyExchangeContractsMember" id="bmifiii_ForwardForeignCurrencyExchangeContractsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="EquitySecuritiesPreferredStocksMember" id="bmifiii_EquitySecuritiesPreferredStocksMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="EquitySecuritiesCommonStocksMember" id="bmifiii_EquitySecuritiesCommonStocksMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="FixedRateLoanInterestsMember" id="bmifiii_FixedRateLoanInterestsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="ConvertibleCorporateBondsAndNotesMember" id="bmifiii_ConvertibleCorporateBondsAndNotesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="FloatingRateLoanInterestsMember" id="bmifiii_FloatingRateLoanInterestsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="CorporateBondsAndNotesMember" id="bmifiii_CorporateBondsAndNotesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="NetOtherAssetsAndLiabilitiesMember" id="bmifiii_NetOtherAssetsAndLiabilitiesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="InvestmentCompaniesMember" id="bmifiii_InvestmentCompaniesMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="C000000793Member" id="bmifiii_C000000793Member" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="C000000792Member" id="bmifiii_C000000792Member" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
	<element name="C000000790Member" id="bmifiii_C000000790Member" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" abstract="true" nillable="true" xbrli:periodType="duration"/>
</schema>
