# Tagged Questions

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

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### For $(a,b)$, if $m^* ((a,b)) = m^* ( (a,b) \cap E ) + m^*( (a,b) \cap - E)$ then $E$ in $\mathbb{R}$ is measurable

If $m^* ((a,b)) = m^* ( (a,b) \cap E ) + m^*( (a,b) - E)$ for all open intervals $(a,b)$, then $E$ in $\mathbb{R}$ is measurable. How do I prove this? Totally stuck.
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### Continuous, strictly increasing function that maps a set of positive (lebesgue) measure onto a set of measure zero?

Is there a continuous, strictly increasing (real-valued) function on the interval $[0,1]$ that maps some set of positive (lebesgue) measure onto a set of measure zero? Should I play with cantor ...
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### Lebesgue integrability and measurable functions

Let $f$ be a nonnegative function on the reals. What does the (Lebesgue) measurability of $f$ have to do with the (Lebesgue) integrability of $\int f$? I've spent some time studying the definition at ...
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### If $u \in L^2(\Omega)$, then $\text{sign}u \in L^2(\Omega)$?

If $\Omega$ is a bounded domain and $u$ is in $L^2$, why is $\text{sign}(u) \in L^2?$ I am only stuck with the measurabilituy part. the integral is obviously finite on a bounded domain.
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### Continuity at $x$ of increasing function if certain sequences exist

I'm working through the first few chapters of Royden-Fitzpatrick to learn measure theory and I got stuck on this question. Let $f$ be increasing on $I$, an open interval. Then for $x \in I$, $f$ is ...
How come if f is measurable, we might have $|f|^2\neq f^2$? Can you provide an example? I think it is true if f is real.