Does this integrand satisfy assumptions of Tonelli's theorem?

Question

Because I'm not familiar with Lebesgue integral, please tell me.

I'm worried whether this equality holds: $$ \int_{0}^{\infty}\int_{0}^{1}\frac{ye^{(1-2/x)y}I_{0}(y)f(x)}{x^{3}}dxdy=\int_{0}^{1}\int_{0}^{\infty}\frac{ye^{(1-2/x)y}I_{0}(y)f(x)}{x^{3}}dydx, $$ where $I_{0}$ is the $0$-Bessel function of the first kind and $f$ is non-negative continuous function on $[0,1]$

My question: Does this equality hold?

I know I should use Tonelli's theorem but I don't prove well that this integrand satisfies assumptions of Tonelli's theorem.

Thank you in advance.

Answer 1

1. The sets $(0,1) $ and $(0,\infty ) $ (equipped with Lebesgue measurw) are sigma finite measure spaces (why?).

2. Every continuous function $f : (0,1)\times (0,\infty) \to \Bbb{R} $ is measurable with respect to the product sigma algebra (which coincides with the Borel sigma algebra). In particular, this holds for your integrand. Here, I assume that $I_0$ is continous (I don't know the definition :) ).

3. Since $I_0$ is nonnegative, so is your integrand (why?).

These are all assumptions needed.