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With the counting measure on $\mathbb{N}$, and the Lebesgue measure on $\mathbb{R}$, consider their product measure space and the function:

$f(x,n) = \begin{cases} -2^n & \text{ if $x\in[2^{-n}, 2^{-n+1})$ } \\ 2^n & \text{ if $x\in [0, 2^{-n})$} \\ 0 & \text {otherwise} \end{cases}$

I have a problem understanding how to compute the two iterated integrals. Could someone show me how to do one of them?

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For each $n$, $x\mapsto f(x,n)$ is a simple function and: $$\int_{\Bbb R}f(x,n)\,dx=2^n(2^{-n}-0)-2^n(2^{-n+1}-2^{-n})=\cdots$$

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