I am reading Measure Theory by Halmos, and I was wondering if someone could help me on the following definition:
If there exists a mean fundamental sequence $\left\{f_{n}\right\}$ of integrable simple functions which converges in measure to an a.e. finite valued, measurable function $f$ on a measure space $(X, \mathbf{S}, \mu).$ Then the $\int f d \mu=\lim _{n} \int f_{n} d \mu$ is defined as the integral of f.
A sequence {fn}is mean fundamental sequence if $\int |f_n-f_m| \rightarrow 0$ as n,m goes to infinity. It Converge in measure to f if for each epsilon, $\mu (\{ x:|f_n(x)-f(x)|> \epsilon \}) \rightarrow 0$ as n goes to inifinity.
I understand that it is the definition, but I was wondering why does $\lim _{n} \int f_{n} d \mu$ exists in the first place(why does this limit exist)?