# Tagged Questions

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### Question about $L^1$-$L^2$ integrable functions

Can somebody tell me what's wrong with the following argument? If $f$ is $L^1$ Lebesgue-integrable, say $f$ positive, then it is bounded almost everywhere by some bound $M$. Then $f^2 < M\cdot f$ ...
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### Prove that this function is measurable

I cannot write a neat proof of this result, so I would like to see how to be precise in these kinds of arguments.. Here is the problem Let $I=[0,1]$ and let $f\colon I\times\mathbb R\to \mathbb R$ ...
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### lebesgue measure of $\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, 0 \leq y \leq 10, z \in \mathbb Z \}$

Find the lebesgue measure of the set: $$\Bigl\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, \quad 0 \leq y \leq 10, \quad z \in \mathbb Z \Bigr\}$$ I think is a null set but for some reason I ...
How do I use this the following result if $f$ is a non-negative measurable function on $X$, then $\int_X f~d\mu =0$ if and only if $f=0$ a.e. on $X.$ to prove that if $f$ be an ...
Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...