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The part of the problem is the next.

Let (X,X,$\mu$) be the measure space on the natural numbers X=$\mathbb{N}$ with the counting measure defined on all subsets of X=$\mathbb{N}$. Let (Y,Y,$\nu$) be an arbitrary measure space.

Show that a set E in Z= X $\times$ Y belongs Z= X $\times$ Y if only if each section $E_{n}$ of E belongs to Y.

I can proof in the this $\Rightarrow$ direction, but I have problems with this $\Leftarrow$ direction, so I think that for this part I need to use the measure space (X,X,$\mu$).

Could you please give some suggestion?

Alex Pozo
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1 Answers1

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Notice that for the counting measure on $X=\mathbb{N}$ the $\sigma$-algebra is $\textbf{X}=\mathcal{P}(\mathbb{N})$. So, if each section $E_n$ belongs to $\mathbb{Y}$, we have that the set $\{n\}\times E_n\in \textbf{Z}=\textbf{X}\times\textbf{Y}$, and $$E=\bigcup_{n\in\mathbb{N}}(\{n\}\times E_n)$$ belongs to $\textbf{Z}$ because it is a countable union of elements in $\textbf{Z}$.

Darío G
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