# Tagged Questions

For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

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### Expectation defined as Riemann integral

I have a question related to the expectation of a continuous random variable and its Riemann integral definition. Consider a continuous real-valued random variable $X$ defined on the probability space ...
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### Show that this integral is finite: $\int_{\mathbb R^3} e^{-\| x\|^2} e^{- a \| x\| \coth (\| x\|) -\| x\| } \, dx$

Haw to prove that the following integral $$\int_{\mathbb R^3} e^{-\| x\|^2} e^{- a \| x\| \coth (\| x\|) -\| x\| } \, dx$$ is finite ? where $a>0$. thanks you in advance
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### derivative of lebesgue integrable function

Suppose we have $f \in L(I)$ and derivative $f'$ exists almost everywhere . It is $f'$ measurable ? I have no idea how to begin to construct the proof .
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### show that $f_h \in L^1(R)$ and $\lim_{h \to 0} f_h(x) = f(x)$ in $L^1(R)$

If $f \in L^1(R)$ and set $f_h(x)= \frac{1}{2h} \int_{x-h}^{x+h}f(t)dt, h>0$ then show that $f_h \in L^1(R)$ and $\lim_{h \to 0} f_h(x) = f(x)$ in $L^1(R)$ To prove f_h is integrable ...