Question on Fubini's Theorem and $\;L^2(a,b)\;$ space Let Hilbert space $\;L^2(a,b)\;$ and $\;k \in L^2([a,b]\times[a,b])\;$. Is it true to claim that:

$\;\int_{a}^b \int_{a}^b {\vert k(t,s) \vert}^2 \;dtds=\int_{a}^b
 \int_{a}^b {\vert k(t,s) \vert}^2 \;dsdt\;$?

In addition, under which circumstances 

$\;\int_{a}^b \int_{a}^b f(t)k(t,s)g(s) \;dtds=\;\int_{a}^b \int_{a}^b f(t)k(t,s)g(s)
 \;dsdt\;$ if $\;f,g\in L^2(a,b)\;$?

I believe the above is true if $\;f,g\in C[a,b]\;$ and $\;k \in C([a,b]\times[a,b])\;$ but I wonder if I could replace continuity by something "less".
EDIT: My initial purpose was to compute the adjoint of $\;T:L^2(a,b)\rightarrow L^2(a,b)\;$ where $\;Tf(s)=\int_{a}^b k(s,t)f(t) \;dt\;$ for $\;f\in L^2(a,b)\;$. In order to achieve that , taking the inner product in $\;L^2\;$ I computed $\;\langle Tf,g \rangle=\int_{a}^b \int_{a}^b f(t)k(t,s)\;dt\;\overline {g(s)} \;ds$. 
Any help would be valuable. Thanks in advance!!
 A: The Fubini-Tonelli Theorem as presented by Folland gives two cases in which
$$\iint f(x,y) \, d(\mu\otimes\nu)(x,y) = \int \left(\int f(x,y) \, d\mu(x) \right) d\nu(y) = \int \left( \int f(x,y) \, d\nu(y) \right) d\mu(x)$$
for a function $f : X \times Y \to \mathbb C$ where $(X, \mu)$ and $(Y, \nu)$ are two measure spaces:


*

*$f \in L^+(X \times Y),$

*$f \in L^1(X \times Y).$


The first case applies in your first case.
For your second case, first we note that since $f,g \in L^2([a,b])$ we have $f \otimes g \in L^2([a,b] \times [a,b]).$
Now we also have $k \in L^2([a,b] \times [a,b])$ so Hölder's inequality ($\|fg\|_1 \leq \|f\|_p \|g\|_q$ when $\frac1p + \frac1q = 1$) implies that $(f \otimes g)\, k \in L^1([a,b] \times [a,b]).$ Thus we here can apply the second case of the Fubini-Tonelli Theorem.
A: Your first statement is of course true.
As for your second statement, it seems your talking about integral operators. What you're looking for holds for general measure spaces $(X, \mu)$ and Hilbert spaces $L^2(\mu)$. That is if $k: X \times X \to \mathbb{C}$ is a measurable function for which there exists constants $c_1,c_2 \geq 0$ such that
$$
\begin{align*}
\int_X \lvert k(x,y) \rvert d\mu(y) & \leq c_1  & \text{a.e.} \\
\int_X \lvert k(x,y) \rvert d\mu(x) & \leq c_2 & \text{a.e.}
\end{align*}
$$
Note that you can define a related operator on $L^2(\mu)$ by
$$
(Kf)(x) = \int_X k(x,y) f(y) d\mu(y).
$$
$K$ is then a bounded operator with $k$ its kernel. All the statements I mention here are not so hard to prove. Also if $k \in C([a,b] \times [a,b])$, it will of course satisfy these conditions since it is a bounded function. 
Edit:
To answer your updated question:
There is the Fubini-Tonelli theorem, that says if
$$
\int_{[a,b]}\left(  \int_{[a,b]} \lvert f(x,y) \rvert d\mu(x) \right) d\mu(y) < +\infty
$$
the function $f$ is integrable and the order of integration may be switched. Working under the assumptions for $k$ that I mentioned, this clearly holds.
