I'm reading Theorem 6.19 in textbook Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland. The proof given by the author is very sketchy. I also change the original statement of part (ii) to make it more concise.
Could you verify if my proof and my formulation of part (ii) are correct?
In part (ii), the case $\infty$ is obtained by the monotonicity of integral. Why don't we do the same for part (i)?
Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be $\sigma$-finite measure spaces, and $(E, \| \cdot \|)$ a Banach space.
$\| \cdot \|_p$ is the $L_p$-norm.
$\lambda = \mu \otimes \nu$ the product measure of $\mu$ and $\nu$.
$f: X \times Y \to E$ is $\lambda$-measurable.
(i) If $1 \le p < \infty$ and $E = \mathbb R_+$, then $$\left[ \int_X \left( \int_Y f(x, y) \mathrm d \nu(y) \right )^{p} \mathrm d \mu(x) \right]^{1 / p} \leq \int_Y \left [ \int_X f^p (x, y) \mathrm d \mu(x) \right]^{1 / p} \mathrm d \nu(y).$$
(ii) If $1 \le p \le \infty$ and $f (x, \cdot)$ is $\nu$-integrable for $\mu$-a.e. $x \in X$, then $$\left [ \int_X \left \| \int_Y f(x, y) \mathrm d \nu(y)\right\|^{p} \mathrm d \mu(x)\right]^{1 / p} \leq \int_Y \left [ \int_X \|f(x, y)\|^{p} \mathrm d \mu(x) \right]^{1 / p} \mathrm d \nu(y).$$
Proof: We start with (i). Let $R$ be the RHS and $L'_q(X, \mu, \mathbb R_+) \triangleq \{g \in L_q(X, \mu, \mathbb R_+) \mid \|g\|_q = 1\}$. The case $p=1$ is true by Tonelli's theorem. For $p >1$, let $q$ be its Hölder's conjugate and $h:X \to \mathbb R, x \mapsto \int_Y f(x, y) \mathrm d \nu(y)$.
First, we consider a function $\varphi: L'_q(X, \mu, \mathbb R_+) \to \overline{\mathbb R}_+, \, g \mapsto \int_X h g \mathrm d \mu$. By Hölder's inequality, $$\sup_{g \in L'_q(X, \mu, \mathbb R_+)} \varphi(g) \le \sup_{g \in L'_q(X, \mu, \mathbb R_+)} \|h\|_p \cdot \|g\|_q = \|h\|_p.$$
Because $h$ is non-negative and measurable, there is a non-decreasing sequence $(h_n)$ of non-negative simple functions converging $\mu$-a.e. to $h$. We define a sequence $(g_n)$ by $$g_n \triangleq \frac{h^{p-1}_n}{\|h_n\|_p^{p-1}}, \quad n \in \mathbb N.$$
It follows from $(p-1)q = p$ that $\|g_n\|_q =1$ and $\|h_n\|_p = \int_X h_n g_n \mathrm d \mu$. This means $g_n \in L'_q(X, \mu, \mathbb R_+)$ for all $n$. As such, \begin{align} \|h\|_p &= \lim_n \|h_n\|_p &&= \lim_n \int_X h_n g_n \mathrm d \mu \\ &\le \lim_n \varphi (g_n) && \le \sup_{g \in L'_q(X, \mu, \mathbb R_+)} \varphi(g). \end{align}
It follows that $\sup_{g \in L'_q(X, \mu, \mathbb R_+)} \varphi(g) = \|h\|_p$. For all $g \in L'_0(X, \mu, \mathbb R_+)$, \begin{align} \varphi (g) =& \int_Y \left ( \int_X f(x, y) g(x) \mathrm d \mu (x) \right ) \mathrm d \nu (y) \quad \text{by Tonelli's theorem} \\ \le& \int_Y \left ( \int_X f^p(x, y) \mathrm d \mu (x) \right )^{1/p} \left ( \int_X g^q(x) \mathrm d \mu (x) \right )^{1/q} \mathrm d \nu (y) \quad \text{by Hölder's inequality} \\ =& R. \end{align} Hence $R$ is an upper bound of $\varphi$ and thus $\|h\|_p \le R$. This completes the proof of (i). Now we're going to prove (ii). First, we consider the case $1 \le p < \infty$. Let's denote the LHS by $L$. Notice that $\|f\|$ is non-negative and measurable. We have \begin{align} L &\le \left [ \int_X \left ( \int_Y \| f(x, y) \| \mathrm d \nu(y)\right )^{p} \mathrm d \mu(x)\right]^{1 / p} \quad \text{because} \quad \left \|\int f \right \|^p \le \left | \int \|f\| \right |^p = \left ( \int \|f\| \right )^p\\ &\le \int_Y \left [ \int_X \|f(x, y)\|^{p} \mathrm d \mu(x) \right]^{1 / p} \mathrm d \nu(y) \quad \text{by (i)}. \end{align}
The case $p =\infty$ is then obtained by the monotonicity of integral.
Note that $\mathbb{R}_+$ is not a Banach space.
One place where your proof breaks down is in defining $g_n$. It is possible that $\lVert h_n \rVert_{L^p} = \infty$ for some $n$, so $g_n$ is not well defined. You need your $h_n$ to be in $L^p(X, \mu)$. The way Folland achieves this in his proof of theorem 6.14 is by using the $\sigma$-finite hypothesis on $X$: Take $E_n \subset X$ of finite measure such that $E_n \nearrow X$, and set $h_n' = h_n\chi_{E_n}$. Then $h_n'$ are simple, $h_n' \nearrow h$, and $h_n' \in L^{p}(X, \mu)$. I think the rest of your proof for (i) goes through.
Basically for $\sigma$-finite $(X, F, \mu)$, for measurable $h \geq 0$ we have $\lVert h \rVert_{L^p} = \sup\{\int_{X}hg\,d\mu : g \geq 0, \lVert g \rVert_{L^q} = 1\}$. You proved this for $p \in [1, \infty)$. A simple argument (which also uses $\sigma$-finiteness of $\mu$) proves it for $p = \infty$. Then the proof of the actual estimate is easy, and it works for all $p \in [1, \infty]$ since Holder's inequality holds for all such $p$. But this only works when $y \mapsto \lVert f(., y) \rVert_{L^p}$ is measurable. For $p < \infty$, this is a consequence of Tonelli's theorem, but I don't think it necessarily holds for $p = \infty$. So your statement for (ii) needs to be careful here.
I agree with you that trying to use monotonicity for $p = \infty$ is not as obvious as it looks because we only have $f(x, y) \leq \lVert f(., y) \rVert_{L^{\infty}}$ for a.e. $x \in X$, when we need it for a.e. $y \in Y$ for a.e. $x \in X$.