Suppose $f(x,y)$ is a Borel function on $\mathbb{R}^2$ which is in the $L^2$-space with respect to the $\mathcal{L}\times\mathcal{L}$. Prove the following: Given any finite rectangle $R=[a,b]\times[a,b]\subset\mathbb{R}^2$, we have $\int_{[c,d]}|f(x,y)|d\mathcal{L}(y)<\infty$ for $\mathcal{L}$-almost all $x\in [a,b]$.
We have $\infty>\int_{\mathbb{R}^2}|f(x,y)|^2d\mathcal{L(x,y)}\geq \int_{R}|f(x,y)|^2d\mathcal{L(x,y)}$. So for $\mathcal{L}$-almost all $x\in[a,b]$, we have $\int_{[c,d]}|f(x,y)|^2d\mathcal{L}(y)<\infty$. Now what should I do? Thanks!
We can use Hölder's inequality (which in the case that is applied here is also known as the Cauchy-Schwarz inequality): $$\|\varphi\psi\|_1\leq \|\varphi\|_p\|\psi\|_q$$ In this case try $\varphi= (y\mapsto f(x,y))$, $\psi=\mathbf 1_{[c,d]}$ and $p=q=2$ to find that $$\int\limits_{[c,d]} |f(x,y)|\,\mathrm dy=\|\varphi\psi\|_1\leqslant\|\varphi\|_2\|\psi\|_2=\left(\int\limits_{[c,d]} |f(x,y)|^2\,\mathrm dy\right)^{1/2} \cdot (d-c)^{1/2}, $$ so $\int_{[c,d]} |f(x,y)|\,\mathrm dy $ is finite for (at least) every $x$ for which $\int\limits_{[c,d]}|f(x,y)|^2\,\mathrm dy$.
The argument above can be generalised to show that if $X$ is a measure space and $\mu$ a finite measure on it, then $L^p(X)\subseteq L^{p'}(X)$ when $p\geqslant p'$ and this inclusion is continuous.