Theorem (Fubini-Tonelli)

Let $(X,\mathscr{M},\mu),(Y,\mathscr{N},\nu )$ be σ-finite measure spaces and let $f:X\times Y\to\mathbb{C}$ be $\mathscr{P}$-measurable.

1. Fubini: If $f:X\times Y\to[0,+\infty]$, set $\varphi:X\to[0,+\infty]$ and $\psi:Y\to[0,+\infty]$ as $\varphi(x)=\int\limits _{Y}f_{x} \, d\nu$ and $\psi(y)=\int\limits _{X}f^{y} \, d\mu$. Then, $\varphi$ is $\mathscr{M}$-measurable and $\psi$ is $\mathscr{N}$-measurable and we have $$\int\limits _{X}\varphi \, d\mu=\int\limits _{Y}\psi \, d\nu$$That is $$\int\limits_{X} \int\limits_{Y}f(x,y) \, d\nu(y) \, d\mu(x) = \int\limits_{Y} \int\limits_{X}f(x,y) \, d\mu(x) \, d\nu(y) =\int\limits _{X\times Y}f \, d(\mu\times \nu)$$
2. Tonelli: Let $\varphi^{*}(x)=\int\limits _{Y}|f_{x}| \, d\nu$. If $\int\limits _{X}\varphi^{*} \, d\mu<\infty$ then $$\int\limits _{X\times Y}|f| \, d(\mu \times \nu)<\infty$$similarly for $\psi^{*}(y)$.
3. If $\int\limits _{X\times Y}|f| \, d(\mu \times \nu)<\infty$ then $f_{x}\in L^{1}(\nu),f^{y}\in L^{1}(\mu)$ and $$\int\limits _{X}\int\limits _{Y}f(x,y) \, d\nu(y) \, d\mu(x)=\int\limits _{Y}\int\limits _{X}f(x,y) \, d\mu(x) \, d\nu(y)$$

Theorem (Fubini-Tonelli)

For $(E,\mathcal{E},\mathbb{P}_{1}),(F,\mathcal{F},\mathbb{P}_{2})$ Probability Spaces and Product Measure $\mathbb{P}$ on $(E\times F,\mathcal{E}\otimes \mathcal{F},\mathbb{P})$. If $f: E\times F\to \mathbb{R}$ is $\mathcal{E}\otimes \mathcal{F}$-measurable and if either $f\ge 0$ or $f\in L^{1}$ then $$\int\limits _{E\times F}f \, d\mathbb{P}=\int\limits _{E}\int\limits _{F}f \, d\mathbb{P_{2}} \, d\mathbb{P}_{1}=\int\limits _{F}\int\limits _{E}f \, d\mathbb{P}_{1} \, d\mathbb{P}_{2}$$with $y\mapsto \int\limits _{E}f_{y}(x) \, d\mathbb{P}_{1}(x)$ measurable.

Theorem (9.4.5)

Suppose $X,Y$ are Independent rvs with Distributions $\mu,\nu$ respectively. Then the distribution of $X+Y$ is given by $\mu*\nu$, where $$(\mu*\nu)(H)=\int\limits _{\mathbb{R}}\mu(H-y) \, \nu(dy) .$$Equivalently, assume $X,Y$ have densities $f,g$, then $X+Y$ has density $f*g$.

Theorem (Fubini-Tonelli)

Let $(X,\mathscr{M},\mu),(Y,\mathscr{N},\nu )$ be σ-finite measure spaces and let $$f:(X\times Y,\mathscr{P}, \mu \times \nu)\to(\mathbb{R},\mathcal{B}(\mathbb{R}))$$be $\mathscr{P}$-measurable. Assume $$\int\limits _{X}\left( \int\limits _{Y}|f(x,y)| \, d\nu(y) \right) \, d\mu(x)<\infty$$(i.e. $f\in L^{2}(X\times Y,\mathscr{P}, \mu \times \nu)$) then, $$\int\limits_{X} \int\limits_{Y}f(x,y) \, d\nu(y) \, d\mu(x) = \int\limits_{Y} \int\limits_{X}f(x,y) \, d\mu(x) \, d\nu(y) =\int\limits _{X\times Y}f \, d(\mu\times \nu)$$