Monotone Convergence Theorem

1. The problem statement, all variables and given/known data



Let f be a non-negative measurable function. Prove that



[itex] \lim _{n \rightarrow \infty} \int (f \wedge n) \rightarrow \int f.[/itex]





3. The attempt at a solution



I feel like I'm supposed to use the monotone convergence theorem.



I don't know if I'm on the right track but I created a sequence of functions so that



[itex]h_1(x) \leq h_2(x) \cdots [/itex] where



[itex]h_1(x) = \min(f_1(x), n)[/itex]



[itex]h_2(x) = \min(f_2(x),n) [/itex]



[itex]\vdots [/itex]



[itex]h_n(x) = \min(f_n(x),n) [/itex]



So the [itex]h(x) = \lim_{n\rightarrow\infty} h_n(x) = \lim_{n\rightarrow \infty} \min(f_n,n) = \lim_{n \rightarrow \infty}\min(f,n) [/itex]