Measurability of function defined by an integral

Question

Let $A$ be a Hilbert-Schmidt operator defined on $L_2[a,b]$ by $$A(\varphi):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$where $K\in L_2([a,b]^2)$. The fact that $A(\varphi)\in L_2[a,b]$ is showed in the proof by Kolmogorov and Fomin (p. 461 here) by the following inequality, where $\psi:=A(\varphi)$:$$\int_{[a,b]}|\psi|^2d\mu\le\|\varphi\|^2\int_{[a,b]}\int_{[a,b]}|K(s,t)|^2d\mu_s d\mu_t$$I would think that, in order to apply a technique by majoration to prove the convergence of the integral of a positive function, we must know that the function is measurable. How do we see that $\psi$ is? I apologise if I ask a trivial question, but my textbook, which is supposed to be an introductory one, often tacitly uses facts never explained before. Thank you so much for any answer!

Answer 1

What you're looking for is Tonelli's Theorem. For your case,

Theorem [Tonelli]: Let $m$ be Lebesgue measure on $[a,b]$; let $\mu$ be the Lebesgue product measure on $[a,b]\times[a,b]$; and let $f$ be a non-negative jointly Lebesgue measurable function on $[a,b]\times[a,b]$. Then, for almost every $x$ the function $f_{x}(y)=f(x,y)$ is Lebesgue measurable on $[a,b]$, and for almost every $y$ the function $f_{y}(x)=f(x,y)$ is Lebesgue measurable on $[a,b]$. Furthermore $\int_{a}^{b}f(x,y)dm(x)$ and $\int_{a}^{b}f(x,y)dm(y)$ are Lebesgue measurable on $[a,b]$. And, $$ \int_{a}^{b}\left[\int_{a}^{b} f(x,y)\,dm(x)\right]dm(y) = \int_{[a,b]\times[a,b]}f\,d\mu = \int_{a}^{b}\left[\int_{a}^{b}f(x,y)\,dm(y)\right]\,dm(x). $$ In particular, one of the three integrals is finite iff the other two are finite and, in all cases, the three are equal.