Convergence in $L^p$ and Minkowski's inequality problem

Question

I am trying to solve the following problem:

Let $(\mathbb R^d,\mathcal M,m)$ with $m$ the Lebesgue measure. Let $f_n \to f$ in $L^p$, with $1 \leq p < \infty$, $g_n \to g$ a.e. and $||g_n||_{\infty} \leq M$ for all $n$. Show that $f_ng_n \to fg$ in $L^p$.

My attempt at a solution was:

We can write $$||f_ng_n-fg||_p=||f_ng_n-fg_n+fg_n-fg||_p$$$$=||(f_n-f)g_n+f(g_n-g)||_p (1)$$

Since $f_n-f \in L^p$ and $g_n \in L^{\infty}$, let $n$ be a natural number and let $M>0$ such that $|g_n(x)|<M$ for all $x$ in $N^c$ for some set $N$ with $m(N)=0$. Then $$\int_{\mathbb R^d}|(f_n-f)g|^pdx=\int_N|(f_n-f)g|^pdx+\int_{N^c}|(f_n-f)g_n|^pdx$$$$\leq M^p\int_{\mathbb R^d}|f_n-f|^pdx<\infty$$

If I could show that $g$ is in $L^{\infty}$ and, moreover, that $g_n \to g$ in $L^{\infty}$, then I could apply Minkowski's inequality in (1) to obtain $$||(f_n-f)g_n+f(g_n-g)||_p \leq ||(f_n-f)g_n||_p+||f(g_n-g)||_p (2)$$

and given all the conditions above, it is easy to show that (2) tends to $0$ when $n$ tends to infinity.

The problem is that I don't know how to show those two properties, as a matter of fact I don't even know if they are actually true. Another thing I was wondering was if it is necessary for two measurable functions $f$ and $h$ to be in $L^p$ in order to apply Minkowski's inequality.

I would appreciate some help. Thanks in advance.

Answer 1

In the term $\|f \, (g - g_n ) \|_p$ you can use dominated convergence theorem: $|f \, (g - g_n)|^p \to 0$ a.e. on $\mathbb{R}^d$ and $2 \, M^p \, |f|^p$ is an integrable majorant. Hence, $$\int_{\mathbb{R}^d} |f \, (g - g_n)|^p \, \mathrm{d}x \to 0.$$

Answer 2

By a variant of Egorov's theorem, given $\epsilon>0$ there exists a Borel set $E\subset\Bbb R^n$ such that $\int_{E^c}|f|^p\,dx<\epsilon$ and $\sup_{x\in E}|g_n(x)-g(x)|\to 0$ as $n\to\infty$. (This is one of Littlewood's three principles: "an a.e. convergent sequence is nearly uniformly convergent".) We then have $$ \int|f|^p|g_n-g|^p\,dx\le (2M)^p\epsilon+\sup_{x\in E}|g_n(x)-g(x)|^p\int_E|f^p|\,dx, $$ which is enough to deal with the second term on the right in your (2).