Let $(X,\mathcal{X},\mu)$ a measure space. Suppose that $(f_{n})$ is a sequence of measurable functions, $f_{n}:X\rightarrow \Bbb{R},$ such that $f_{n}\in L_{p}, 1\leq p < \infty.$ Assume that $f_{n}\rightarrow f$ uniformly, with $f=0$ a.e. or $\frac{1}{f}\in L_{\infty}$. I need to show that $f\in L_{p}$. Also, I need to know if it implies that $f_{n}\rightarrow f$ in $L_{p}.$
My try:
If $f=0$ a.e., so $\int |f|^{p}d\mu=\int_{M}|f|^{p}d\mu+\int_{N}|f|^{p}d\mu$, where $M$ is the set where $f=0$, and $N$ is the set where $f\neq 0$. So, $\mu(N)=0\implies \int|f|^{p}d\mu=\int_{M}|f|^{p}d\mu=\int_{M}0d\mu=0<\infty.$ So, $f\in L_{p}.$
Now, what if $f$ is not $0$ a.e., and so $\frac{1}{f}\in L_{\infty}?$
If that is the case, $\left|\left|\frac{1}{f}\right|\right|_{\infty}<\infty,$ where $$\left|\left|\frac{1}{f} \right|\right |_{\infty}=\inf\left\{\sup\left\{\frac{1}{|f(x)|}:x\notin N\right\}:N\in\mathcal{X},\mu(N)=0\right\}<\infty.$$
Why this implies that $\int |f|^{p}d\mu<\infty$ ? And what can I say about the convergence in $L_{p}?$