Completeness of $L^1$ space
I'm studying this proof:
https://www.math.utah.edu/~savin/L2_5210.pdf
but I can't understand the step when he says: Hence, by the Monotone Convergence Theorem, there exists an integrable function $\phi$, such that $\lim_{n\to \infty}\phi_n(x) =\phi(x)$ for almost all $x$.
The monotone convergence theorem that I have studied does not have the existence of the limit function $\phi$ in the thesis, but in the hypothesis. In fact, why the limit $\lim_{n\to \infty}\phi_n(x)$ could not be infinite on a non-zero measure set?
Solution 1:
Here is a way you can understand this step :
-
$\varphi := \lim_{n \rightarrow +\infty} \varphi_n$ exists since $(\varphi_n)$ is an increasing sequence of function. At this step, we still don't know that $\varphi$ is integrable.
-
By Monotone Convergence Theorem, $$\lim_{n \rightarrow +\infty} \int \varphi_n = \int \varphi$$
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Because you have $\displaystyle{\int \varphi_n \leq \sum ||f_i||}$ for every $n$, then you deduce from 2. that $$\int \varphi \leq \sum ||f_i|| < \infty$$
so $\varphi$ is integrable and you are done.