Lebesgue integral proof, prove integral is zero

[jinsing]

Homework Statement

Suppose S is a set with finite measure, f>=0 on S and f is Lebesgue measurable. For each n in N define E_n = {x \in S | f(x) > n} Prove that \lim_{n \rightarrow \infty} \int_{E_n} f = 0

Homework Equations

Definition of Lebesgue measurability for unbounded functions

The Attempt at a Solution

Honestly, I'm not entirely sure where to even begin with this one. Would I want to somehow incorporate the epsilon definition of limit? Or should I define f_n(x) = f(x) if f(x) <= n and n if f(x) > n, and show that limit will equal 0? Or none of these ideas? Just a little push in the right direction would be really helpful.

Thanks!

 

[mighty2000]

One possible approach to proving this statement is by using the Monotone Convergence Theorem. This theorem states that if a sequence of measurable functions {f_n} converges pointwise to a function f and |f_n| <= g for all n, where g is an integrable function, then the limit of the integrals of f_n is equal to the integral of f.

In this case, we have a sequence of functions {f_n} where f_n(x) = f(x) if f(x) <= n and 0 if f(x) > n. Notice that |f_n| <= f for all n, since f_n(x) is always either equal to f(x) or 0. Also, since f is measurable and S has finite measure, f is integrable on S. Therefore, we can use the Monotone Convergence Theorem to show that the limit of the integrals of f_n is equal to the integral of f.

Now, we need to show that the limit of the integrals of f_n is equal to 0. This can be done by showing that the sequence of integrals is decreasing and bounded below by 0. Since f_n(x) = 0 for all x in E_n, we have that the integral of f_n over E_n is equal to 0 for all n. Also, since f_n(x) <= f(x) for all x in S, we have that the integral of f_n over S is less than or equal to the integral of f over S, which is a finite number. Therefore, the sequence of integrals is decreasing and bounded below by 0, which means that the limit of the integrals must be equal to 0.

Thus, we have shown that the limit of the integrals of E_n is equal to 0, which proves the statement.