A problem with the Lebesgue-Stieltjes measure

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Let $f:mathbbR mapsto mathbbR$ be a nondecreasing function and let $lambda_f$ be the Lebesgue–Stieltjes measure generated by $f$. Prove that $lambda_f(x_0)=0$ if and only if $f$ is left-continuous at $x_0$.










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  • This is an immediate consequence of the definition of the Lebesgue-Stieltjes measure.
    – MathOverview
    Sep 10 at 21:17










  • Could you explain to me in what form? It is an exercise in my task.
    – EuMJsus
    Sep 11 at 2:02














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Let $f:mathbbR mapsto mathbbR$ be a nondecreasing function and let $lambda_f$ be the Lebesgue–Stieltjes measure generated by $f$. Prove that $lambda_f(x_0)=0$ if and only if $f$ is left-continuous at $x_0$.










share|cite|improve this question





















  • This is an immediate consequence of the definition of the Lebesgue-Stieltjes measure.
    – MathOverview
    Sep 10 at 21:17










  • Could you explain to me in what form? It is an exercise in my task.
    – EuMJsus
    Sep 11 at 2:02












up vote
-1
down vote

favorite









up vote
-1
down vote

favorite











Let $f:mathbbR mapsto mathbbR$ be a nondecreasing function and let $lambda_f$ be the Lebesgue–Stieltjes measure generated by $f$. Prove that $lambda_f(x_0)=0$ if and only if $f$ is left-continuous at $x_0$.










share|cite|improve this question













Let $f:mathbbR mapsto mathbbR$ be a nondecreasing function and let $lambda_f$ be the Lebesgue–Stieltjes measure generated by $f$. Prove that $lambda_f(x_0)=0$ if and only if $f$ is left-continuous at $x_0$.







measure-theory lebesgue-measure






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asked Sep 10 at 21:03









EuMJsus

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  • This is an immediate consequence of the definition of the Lebesgue-Stieltjes measure.
    – MathOverview
    Sep 10 at 21:17










  • Could you explain to me in what form? It is an exercise in my task.
    – EuMJsus
    Sep 11 at 2:02
















  • This is an immediate consequence of the definition of the Lebesgue-Stieltjes measure.
    – MathOverview
    Sep 10 at 21:17










  • Could you explain to me in what form? It is an exercise in my task.
    – EuMJsus
    Sep 11 at 2:02















This is an immediate consequence of the definition of the Lebesgue-Stieltjes measure.
– MathOverview
Sep 10 at 21:17




This is an immediate consequence of the definition of the Lebesgue-Stieltjes measure.
– MathOverview
Sep 10 at 21:17












Could you explain to me in what form? It is an exercise in my task.
– EuMJsus
Sep 11 at 2:02




Could you explain to me in what form? It is an exercise in my task.
– EuMJsus
Sep 11 at 2:02















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