Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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Convergence in $L^p$ and convergence almost everywhere

Why $f_n$ converges to $f$ in $L^p$ space implies that exists subsequence of $f_n$ converging to $f$ almost everywhere?
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relation between measurable function and continuous functions

what is the relation between measurable function and continuous function? Which one implies the other? Examples for both type of functions. Whether sinx, cosx, tanx etx are measurable functions on ...
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Real Analysis, Folland Proposition 2.11/Exercise 10 Measurable Functions

Question Proposition 2.11 (Exercise 10) - The following implications are valid if and only if the measure is complete: a.) If $f$ is measurable and $f = g$ $\mu$-a.e., then $g$ is measurable. ...
Notation: $A+B = \{ a + b : a \in A, b \in B \}.$ H. Steinhaus proved the classical result that $A+B$ contains an interval if $A$ and $B$ are both measurable subsets of the real line, each ...