Completeness of $L^p$ space

Question

In proving completeness of $L^p$ space, I got stuck at a step where this situation occurs. Let $\langle\,f_n\,\rangle$ be sequence of measurable functions on $[0,1].$ Can we say that in $L^p$ space, integration of summation of modulus of functions is equal to the summation of integration of modulus of functions. Here we know that each function $f_n$ is measurable. This is generally true in uniform convergence of series . What can we say about this in $L^p$ space. Thanks in advance

Answer 1

In $L^p[0,1]$ there are two relevant theorems. First is Egorov's theorem which says

Let $(f_n)$ be a sequence of measurable functions on a set $A$ of finite measure that converge pointwise a.e. to a function $f$. Let $\varepsilon > 0$. Then there is a measurable subset $B$ of $A$ such that $\mu(B) \ge \mu(A) - \varepsilon$ and $f_n|B$ converges uniformly.

This can be used to prove the "big theorem" which is Lebesgue's Dominated Convergence Theorem which says

Suppose $f_n \to f$ pointwise a.e. such that the sequence is dominated by some integrable function $g$. That is, $|f_n(x)| \le g(x)$ for all $n$ and $x$ and $g$ is integrable. (This condition says, in particular, that $f_n$ is integrable by comparison with $g$.) We can conclude that $f$ is integrable and $$\int f_n \to \int f. $$