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Fubuni's theorem

Statement

Given an integrable function $f: \mathbb{R}^2 \to \mathbb{R}$, then $x \mapsto f(x,y)$ is integrable for almost all y.

Furthermore, we define $F(y) = \int f(x,y)dx$ for almost all $y$, and $F$ is integrable and:

$$\int_{\mathbb{R}^2} f(x,y) d(x,y) = \int_\mathbb{R}(\int_\mathbb{R} f(x,y) dx) dy$$.

and a similar statement holds in the other variable.

all these pages adapted with probably insufficient credit from my university's lecture notes