# Convergence in an $L^p$ space

lebesgue-integrallp-spacesmeasure-theory

Let $$\{f_n\}$$ be a sequence of measurable functions on a domain $$E$$ and $$p$$ be a positive finite real number such that

(a) $$\{f_n\}$$ converges to a measurable function $$f$$ almost everywhere, and;

(b) $$\lim _{n \rightarrow \infty} (||f_n||_p – ||f||_p) = 0.$$

Prove that $$||f_n – f||_p \rightarrow 0.$$

I believe what we have to show is that $$\lim_{n \rightarrow \infty} \int_E |f_n – f|^p = 0.$$

My approach is to hope to invoke the Dominated Convergence theorem, i.e., prove that there exists some $$g \in L^1(E)$$ such that $$|f_n| \leq g$$ for almost everywhere for all $$n$$, then we can have
$$\int_E |f_n – f| \rightarrow 0.$$

Then by (a), we can find an integer $$K$$ such that $$|f_k – f| < 1$$ for all $$k \geq K$$. Then
$$0 \leq \int _E |f_k – f|^p \leq \int _E |f_k – f|.$$

Since the integral on the right approaches $$0$$, we conclude that the same is true for the middle term and we will be done.

However, I have no idea how to obtain the $$g$$ as mentioned before, as well as use condition (b) given in the question. Some help would be appreciated.

This question has been answered many many times for $$p \geq 1$$. For $$0 this follows from Fatou's Lemma: $$|f-f_n|^{p} \leq |f|^{p}+|f_n|^{p}$$ in this case so $$|f|^{p}+|f_n|^{p} -|f-f_n|^{p}$$ is non-negative Fatou's Lemma applied to this sequence gives the result immediately.