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