I am at the moment unfamiliar with the subject of Bochner spaces.
Let us say, for simplicity, that $\Omega=\mathbb{R}^n$. Let us assume also that $1\le p \le \infty$, $1\leq q < \infty.$ Suppose there is a sequence of functions $f_n(x,t)$, with $(x,t)\in \mathbb{R}^n \times [0,T]$ such that the following strong convergence holds:
$$f_n \to f \text{ in } L^q(0,T; L^p(\Omega)).$$
The question is:
Can i find a subsequence $f_{n_k}$ such that
$$f_{n_k}(x,t) \to f(x,t)\text{ pointwise a.e. in }\mathbb{R}^n \times [0,T]?$$
I tried to argue as follows; since
$\Vert(f_n-f)(\cdot,t)\Vert_{L^p(\Omega)}$ converges to 0 in $L^q([0,T])$, we can find a subsequence $f_{n_m}$ such that
$$\Vert(f_{n_m}-f)(\cdot,t)\Vert_{L^p(\Omega)} \to 0\text{ a.e. }t\in [0,T].$$
Now I want to use the fact that $f_{n_m}-f \to 0$ in $L^p(\Omega)$ to claim that I can extract another subsequence so that it converges pointwise a.e. on $\Omega \times [0,T]$. But the problem is that, I believe that the choice of subsequence may vary for each choice of $t\in [0,T]$. Is there a way to proceed, or is my statement false?
Best Answer
The answer is yes, thanks to Kakashi's idea in the comments above of looking at convergence in measure.
Indeed: for $\varepsilon > 0$, consider $E_{n,\varepsilon} := \{(x,t) \in \Omega \times [0,T] \mid |f(x,t) - f_n(x,t)| \geq \varepsilon\}$, and denote by $\lambda$ all the Lebesgue measures.
First, consider the case $q \geq p$.
Then, we get: $$\begin{split}\varepsilon^p \lambda(E_{n,\varepsilon}) &= \int_{E_{n,\varepsilon}} \varepsilon^p \,\mathrm{d}t\mathrm{d}x\\ &\leq \int_{E_{n,\varepsilon}} |f(x,t) - f_n(x,t)|^p \,\mathrm{d}t\mathrm{d}x\\ &\leq \iint_{\Omega \times [0,T]} |f(x,t) - f_n(x,t)|^p \,\mathrm{d}x\mathrm{d}t\end{split}$$
Fubini-Tonelli lets us change the order of integration, and we can apply $x \mapsto x^{\frac{q}{p}}$ to the inequality, hence: $$\left(\varepsilon^p \lambda(E_{n,\varepsilon})\right)^{\frac{q}{p}} \leq \left(\int_{[0,T]} \left(\int_{\Omega}|f(x,t) - f_n(x,t)|^p \,\mathrm{d}x\right)\mathrm{d}t\right)^{\frac{q}{p}}$$
Now we can apply Jensen's inequality on $[0,T]$ of finite measure $T$ with the function $x \mapsto x^\frac{q}{p}$, which is convex due to the assumption that $q \geq p$, which provides: $$\begin{split} 0 \leq \left(\varepsilon^p \lambda(E_{n,\varepsilon})\right)^{\frac{q}{p}} &\leq \left(\int_{[0,T]} \left(\int_{\Omega}|f(x,t) - f_n(x,t)|^p \,\mathrm{d}x\right)\mathrm{d}t\right)^{\frac{q}{p}}\\ &\leq T^{\frac{q}{p}}\left(\frac{1}{T}\int_{[0,T]} \left(\int_{\Omega}|f(x,t) - f_n(x,t)|^p \,\mathrm{d}x\right)\mathrm{d}t\right)^{\frac{q}{p}}\\ &\leq T^{\frac{q}{p} - 1}\int_{[0,T]} \left(\int_{\Omega}|f(x,t) - f_n(x,t)|^p \,\mathrm{d}x\right)^{\frac{q}{p}}\mathrm{d}t\\ &\leq T^{\frac{q}{p} - 1} \|f - f_n\|^q_{L^q(L^p)} \xrightarrow[n \to \infty]{} 0\end{split}$$
This gives us $\lambda(E_{n,\varepsilon}) \to 0$ when $n \to \infty$ for all $\varepsilon > 0$, thus $(f_n)_n$ converges globally in measure to $f$, which means that $(f_n)_n$ admits a subsequence which converges almost everywhere to $f$.
Now, consider the case $q < p$.
Similarly to what was done before, we'll still use Jensen's inequality, but this time "the other way round" with the concave function $x \mapsto x^{\frac{q}{p}}$. However, since that'll be done on the $\Omega$-integral so to speak, we'll need this time to consider local convergence in measure.
Pick $F \subset \Omega \times [0,T]$ of finite measure, and $F_1 \subset \Omega$ of finite measure such that $F \subset F_1 \times [0,T]$. Finally, instead of $E_{n, \varepsilon}$, define $E_{n, \varepsilon, F} := \{(x,t) \in F \mid |f(x,t) - f_n(x,t)| \geq \varepsilon\}$.
Then, we'll have: $$\begin{split}\varepsilon^q \lambda(E_{n,\varepsilon, F}) &= \int_{E_{n,\varepsilon, F}} \varepsilon^q\,\mathrm{d}t\mathrm{d}x\\ &\leq \int_{E_{n,\varepsilon, F}} |f(x,t) - f_n(x,t)|^q \,\mathrm{d}t\mathrm{d}x\\ &\leq \iint_{F_1 \times [0,T]} |f(x,t) - f_n(x,t)|^q \,\mathrm{d}x\mathrm{d}t\\ &\leq \lambda(F_1)\int_{[0,T]} \frac{1}{\lambda(F_1)} \left(\int_{F_1} |f(x,t) - f_n(x,t)|^{p\frac{q}{p}}\,\mathrm{d}x\right)\mathrm{d}t\\ &\overset{\text{Jensen}}{\leq} \lambda(F_1) \int_{[0,T]} \left(\frac{1}{\lambda(F_1)}\int_{F_1} |f(x,t) - f_n(x,t)|^p \,\mathrm{d}x\right)^{\frac{q}{p}}\mathrm{d}t\\ &\leq \lambda(F_1)^{1 - \frac{q}{p}}\|f_n - f\|^q_{L^q(L^p)}\end{split}$$ Thus we conclude in a very similar fashion as for the previous case, as local convergence in measure still grants the existence of the desired subsequence when the measure is $\sigma$-finite, which is the case here.