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## Hot answers tagged measurable-functions

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### Does any measurable function whose integral over any interval is $0$ satisfies $f(x)=0, a.e.$?

In order for $\int_a^b f(x)\; dx$ to be defined (as a Lebesgue integral), we need $f \in L^1(a,b)$. The Lebesgue differentiation theorem then says that for almost every $t$,  f(t) = \lim_{\...
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Define, for each $t \in [0,1]$, $(f(t))^{-1}(\mathscr{B}(\mathbb{R}^d)):=\{\{f\in C[0,1]:f(t)\in B\}|B \in \mathscr{B}(\mathbb{R}^d)\}$ the $\sigma$-algebra generated by the coordinate projection $f \... • 15.8k 2 votes ### Image of measurable sets under one to one (a.e) functions That would depend heavily on the measure. In an extreme case, if your measure$\mu$is concentrated on a singleton then the condition "one-to-one a.e." is satisfied trivially and the ... • 1,853 2 votes ### A conjecture about measurable functions Your conjecture is false. Let$\mathcal H$be the$\sigma$-algebra of Lebesgue measurable sets in$\mathbb R$. Knowing whether each half-line$(-\infty, x)$occurs, i.e. whether$X < x$, uniquely ... • 453k 1 vote Accepted ### equivalence for Borel-measurable function In order for$g$to be$\mathcal{A}$-measurable, the preimage of every$\mathcal{A}$-measurable set has to be in$\mathcal{A}$. The preimage$f^{-1}$of any interval$[a,b]$with$b<0\$ is empty. If ...
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