Let $p:\mathbb{R}^{d}\rightarrow [2,4]$ be a measurable function (we can relax this to $[1,c]$, for any $c>1$), and let $f_{n}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a sequence of measurable functions such that
- $\int_{\mathbb{R}^{d}}|f_{n}(x)|^{p(x)}dx<\infty,\enspace\forall n$
- $\lim_{m,n\rightarrow\infty}\int_{\mathbb{R}^{n}}|f_{n}(x)-f_{m}(x)|^{p(x)}dx=0$
We want to show that there exists a measurable function $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ such that
- $\int_{\mathbb{R}^{d}}|f(x)|^{p(x)}dx<\infty$
- $\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{d}}|f_{n}(x)-f(x)|^{p(x)}dx=0$
My original idea is as follows (skip to the bottom if you just want the question). For any $0<\epsilon\leq 1$, $$\epsilon^{4}|\{|f_{n}-f_{m}|\geq\epsilon\}|\leq\int_{\mathbb{R}^{d}}|f_{n}(x)-f_{m}(x)|^{p(x)}dx\rightarrow 0, \enspace n,m\rightarrow\infty$$ So $\{f_{n}\}$ is Cauchy in measure, hence there exists a subsequence $\{f_{n_{k}}\}$ such that $f_{n_{k}}\rightarrow f$ a.e. By Fatou's lemma, $$\int_{\mathbb{R}^{d}}|f_{n}(x)-f(x)|^{p(x)}dx\leq\liminf_{k\rightarrow\infty}\int_{\mathbb{R}^{d}}|f_{n}(x)-f_{n_{k}}(x)|^{p(x)}dx\leq\epsilon$$, given $\epsilon>0$, for all $n$ sufficiently large. Whence, $$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{d}}|f_{n}(x)-f(x)|^{p(x)}dx=0$$ By convexity, $$|f(x)|^{p(x)}\leq 2^{p(x)-1}[|f_{1}(x)-f(x)|^{p(x)}+|f_{1}(x)|^{p(x)}]\leq 2^{3}[|f_{1}(x)-f(x)|^{p(x)}+|f_{1}(x)|^{p(x)}]$$ whence $$\int_{\mathbb{R}^{d}}|f(x)|^{p(x)}dx\lesssim\int_{\mathbb{R}^{d}}|f_{1}(x)-f(x)|^{p(x)}dx+\int_{\mathbb{R}^{d}}|f_{1}(x)|^{p(x)}dx<\infty$$
I am interested in a way to prove the above result by approximating $p(x)$ by simple functions, as this was the suggested argument in the original prelim problem.
If $p(x)=\sum_{k=1}^{l}p_{k}\mathbf{1}_{E_{k}}$, where the $E_{k}$ are disjoint, then applying the ordinary Riesz-Fischer theorem on each $E_{i}$ with the appropriate exponent gives us the desired $f$. For general $p$, it is not clear to me how to approximate $p$ by simple functions. Specifically, we want to have an estimate like $$\int_{E_{i}}|f_{n}(x)-f_{m}(x)|^{p_{i}}dx\leq\int_{E_{i}}|f_{n}(x)-f_{m}(x)|^{p(x)}dx$$ If we approximate $p(x)$ by its values on level sets $\{a_{i}\leq |p(x)|<b_{i}\}$, then whether we use $a_{i}$ or $b_{i}$ depends on the value of $|f_{n}(x)-f_{m}(x)|$. Any thoughts?