Proving the Monotone Convergence Theorem for Non-Negative Measurable Functions

[BrainHurts]

Homework Statement

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

\lim _{n \rightarrow \infty} \int (f \wedge n) \rightarrow \int f.

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

h_1(x) \leq h_2(x) \cdots where

h_1(x) = \min(f_1(x), n)

h_2(x) = \min(f_2(x),n)

\vdots

h_n(x) = \min(f_n(x),n)

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

 

[Dick]

Homework Statement

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

\lim _{n \rightarrow \infty} \int (f \wedge n) \rightarrow \int f.

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

h_1(x) \leq h_2(x) \cdots where

h_1(x) = \min(f_1(x), n)

h_2(x) = \min(f_2(x),n)

\vdots

h_n(x) = \min(f_n(x),n)

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

I'm guessing $f \wedge n$ means min(f,n)? Then why not just define $f_n=\min(f,n)$? What do YOU mean by your $f_n$??

 

[BrainHurts]

I'm guessing $f \wedge n$ means min(f,n)? Then why not just define $f_n=min(f,n)$? What do YOU mean by your $f_n$??

yes f \wedge n = \min(f, n)

so define

f_1(x) = \min(f(x),1)

f_2(x) = \min(f(x),2)

\vdots

f_n(x) = \min(f(x),n)

In short we still have that f_1(x) \leq f_2(x) \leq \cdots \leq f_n(x) for all x.

Well I want \lim_{n \rightarrow \infty}f_n \rightarrow f for all x because if I have this situation I can use the monotone convergence theorem.

I'm just not sure if that's the case.

 

[Dick]

yes f \wedge n = \min(f, n)

so define

f_1(x) = \min(f_1(x),1)

f_2(x) = \min(f_2(x),2)

\vdots

f_n(x) = \min(f_n(x),n)

In short I believe we still have that f_1(x) \leq f_2(x) \leq \cdots \leq f_n(x) for all x.

Well I want \lim_{n \rightarrow \infty}(f_n \wedge n) \rightarrow f for all x because if I have this situation I can use the monotone convergence theorem.

I'm just not sure if that's the case.

I think it's all fine, except $f_1(x) = \min(f_1(x),1)$ doesn't do a good job of defining $f_1$. $f_1(x) = \min(f(x),1)$ is much better. Are you asking why $f_n \rightarrow f$?

 

[BrainHurts]

Sorry I just made some edits, I saw my mistake.

 

[BrainHurts]

I think I'm covered, because I'll have to consider the case when f is finite and f is infinite correct?

 

[Dick]

I think I'm covered, because I'll have to consider the case when f is finite and f is infinite correct?

Maybe. f is nonnegative. So either the integral exists or it's '+infinity'. I think in either case the sequence convergence is correct. If you allow things like '+infinity'. Think about a function like f(x)=1/x^2. Define f(0) however you want.