Precise Defintion of Fubini-Tonelli's Theorm

Question

I have been getting confused by multiple source's versions of Fubini-Tonelli's theorem and I would like to simply make sure I am getting the definition straight.

It seems to me that Fubini-Tonelli's theorem has 2 versions:

1. If $\Omega = \Omega_1 \times \Omega_2$, $F = F_1 \times F_2$ and $\mu = \mu_1 \times \mu_2$ where $\mu_1,\mu_2$ are sigma finite measures, $f$ is measurable with respect to $(\Omega, F)$ and $f$ is integrable with respect to $(\Omega, F)$, then:

   $$\int_\Omega f(x,y) d(\mu_1 \times \mu_2) = \int_{\Omega_1} \int_{\Omega_2} f(x,y) d \mu_1 d \mu_2 = \int_{\Omega_1} \int_{\Omega_2} f(x,y) d \mu_2 d \mu_1 $$

2. If $\Omega = \Omega_1 \times \Omega_2$, $F = F_1 \times F_2$ and $\mu = \mu_1 \times \mu_2$ where $\mu_1,\mu_2$ are sigma finite measures, $f$ is measurable with respect to $(\Omega, F)$ and if either $\int_{\Omega_1} \int_{\Omega_2} |f(x,y)| d \mu_1 d \mu_2 < \infty$ or $ \int_{\Omega_1} \int_{\Omega_2} |f(x,y)| d \mu_2 d \mu_1 < \infty$, then:

   $$\int_\Omega f(x,y) d(\mu_1 \times \mu_2) = \int_{\Omega_1} \int_{\Omega_2} f(x,y) d \mu_1 d \mu_2 = \int_{\Omega_1} \int_{\Omega_2} f(x,y) d \mu_2 d \mu_1 $$

Is this correct? Are there any conditions I have missed?

Answer 1

The two "versions" of the theorem can be seen as the two sides of an "if and only if". Namely, you can write what you wrote in the following way:

If $\Omega=\Omega_1\times\Omega_2$, $F=F_1\times F_2$ and $μ=μ_1\times μ_2$, where $μ_1,μ_2$ are sigma-finite measures, and $f$ is measurable with respect to $(Ω,F)$, the following statements are equivalent:

• $\displaystyle \int_\Omega |f(x,y)| d(\mu_1 \times \mu_2)<\infty$

• $\displaystyle \int_{\Omega_1} \int_{\Omega_2} |f(x,y)| d \mu_1 d \mu_2 < \infty$

• $\displaystyle \int_{\Omega_1} \int_{\Omega_2} |f(x,y)| d \mu_2 d \mu_1 < \infty$

When the above conditions hold, we have $$ \int_\Omega f(x,y) d(\mu_1 \times \mu_2) = \int_{\Omega_1} \int_{\Omega_2} f(x,y) d \mu_1 d \mu_2 = \int_{\Omega_1} \int_{\Omega_2} f(x,y) d \mu_2 d \mu_1. $$