# Questions tagged [almost-everywhere]

For questions about the concept of "almost everywhere", that is, questions about properties which holds everywhere, except on a set of measure 0. This is involved in probability theory as well as in the case of infinite measure space. Use the appropriate tags to specify the context.

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### Is the following characterisation of measurable functions true?

I am self studying measure theory.I measure theory it is often important to check if a function is measurable.If the function is continuous then it is measurable of course.But if the function is not ...
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There are a couple different equivalent definitions of almost sure (a.s.) convergence: $\forall \varepsilon>0\quad P(\liminf_{ n\uparrow \infty}\{\|X_n-X\|\leq\varepsilon\})=1$ $\mathbb{P}(\{\... 0 votes 1 answer 41 views ### Proof verification of functions being the same almost everywhere Let ($X, \mathscr{F}, \mu)$be a measure space and,$T:X \to X\mathscr{F}/\mathscr{F}$measurable map such that:$\mu(C)=\mu(T^{-1}(C))$for all$C\in \mathscr{F}$, and let$u\in\mathscr{L}^1(\mu).... 52 views

### L2 convergence for a simple function approximation

Consider the problem on the picture. I am struggling with part (b) of the excercise. I have managed to show that we have $L^1$ convergence, but I am unable to show $L^2$ convergence. Does anyone have ...
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### A.e. $L^p$ convergence implies a.e. convergence along sub-sequence
The book on Bochner spaces that I am currently looking at contains the following theorem: Let $\tilde{u}:[0,T] \to L^p(a,b)$ be Bochner measurable for some $1\leq p< \infty$. Define u: [0,T] \...