Let $\sigma: [ 0,\infty) \rightarrow [0,\infty]$ be a Borel-measurable, nonnegative, extended-real valued function, when is the Lebesgue integral $\int_{0}^{t}\sigma(u) du$ a continuous function of t? Thanks!
When is the Lebesgue integral continuous?
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0Note that any antiderivative of a locally integrable function is not merely continuous, but also absolutely continuous. – 2012-07-28
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0Do we know something else, like $\sigma\in L^1_{loc}[0,\infty)$ or something? – 2012-07-28
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0If $\sigma$ is locally integrable, then the function is in fact differentiable a.e. (ie, the Lebesgue points). – 2012-07-28
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0Notice that $\sigma$ is nondecreasing, which implies that the set $\{t: \sigma(t)<\infty \}$ is an interval. In this interval (possibly empty) you have local integrability, hence local absolute continuity of the antiderivative. Outside of it, the antiderivative is $\equiv \infty$. – 2012-07-28
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0@Leonid: No assumption on the variations of $\sigma$. – 2012-07-28
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0@did Sorry, I meant the unnamed antiderivative of $\sigma$. – 2012-07-28
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0@Leonid: Yep, that was my guess... :-) – 2012-07-28
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0@LeonidKovalev Thank you! That is the answer I need. – 2012-07-30
1 Answers
Since my answer-comment had a couple of typos and did not cover all bases, I'm posting an actual answer in the answer box (what a novel idea!).
Let $F(t)=\int_0^{t}\sigma(u)\,du$, which is a well-defined function taking values in $[0,\infty]$. Since $\sigma\ge 0$, $F$ is nondecreasing, which implies that the set $I=\{t:F(t)<\infty\}$ is an interval. On this (possibly empty) interval $\sigma$ is locally integrable, and therefore $F$ is locally absolute continuous. Outside of the interval $F\equiv +\infty$.
The question remains: what happens at right endpoint of $I$ when this endpoint is finite? Here a discontinuity may occur, e.g., if $\sigma\equiv \infty$ or $\sigma=\infty \chi_{[1,\infty)}$ or, to give a finite example, $\sigma(u)=(u-1)^{-1}\chi_{(1,\infty)}$. Denoting the right endpoint of $I$ by $b$, we can formulate the criterion for continuity at $b$ (hence on the entire line) as $\int_0^b \sigma(u)\,du=\infty$.
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0Thank you for this answer, it is very clear. – 2012-08-18