# Fubini and Tonelli For $$f(x, y)$$ non-negative and measurable, for almost every $$x\in {\mathbb{R}}^n$$,

• $$f_x(y)$$ is a measurable function
• $$F(x) = \int f(x, y) ~dy$$ is a measurable function,
• For $$E$$ measurable, the slices $$E_x \coloneqq\left\{{y {~\mathrel{\Big\vert}~}(x, y) \in E}\right\}$$ are measurable.
• $$\int f = \int \int F$$, i.e. any iterated integral is equal to the original.

For $$f(x, y)$$ integrable, for almost every $$x\in {\mathbb{R}}^n$$,

• $$f_x(y)$$ is an integrable function
• $$F(x) \coloneqq\int f(x, y) ~dy$$ is an integrable function,
• For $$E$$ measurable, the slices $$E_x \coloneqq\left\{{y {~\mathrel{\Big\vert}~}(x, y) \in E}\right\}$$ are measurable.
• $$\int f = \int \int f(x,y)$$, i.e. any iterated integral is equal to the original

If any iterated integral is absolutely integrable, i.e. $$\int \int {\left\lvert {f(x, y)} \right\rvert} < \infty$$, then $$f$$ is integrable and $$\int f$$ equals any iterated integral.

\begin{align*} F(x):=\int_{0}^{x} f(y) d y \quad \text { and } \quad G(x):=\int_{0}^{x} g(y) d y \\ \implies \int_{0}^{1} F(x) g(x) d x=F(1) G(1)-\int_{0}^{1} f(x) G(x) d x .\end{align*}

Fubini-Tonelli, and sketch region to change integration bounds. #todo

#todo