# Rudin Remark 11.23

[SOLVED] Rudin Remark 11.23

## Homework Statement

Prove: If f is measurable and bounded on E, and if $\mu(E) < + \infty$, then f is Lebesgue integrable on E.

## Homework Equations

Here is how Rudin defines the Lebesgue integral. Everyone probably already knows this but anyway:

Suppose
(51) $$s(x) = \sum_{i=1}^n c_i K_{E_i} (x) \mbox{ (for x \in X, c_i >0)}$$

(where K is the indicator function) is measurable, and suppose E is a measurable set. We define

(52) $$I_E (s) = \sum_{i=1}^n c_i \mu(E \cap E_i)$$

If f is measurab;e and nonnegative, we define the Lebesgue integral of f over the set E as

(53) $$\int_E f d\mu = \sup I_E (s)$$

where the sup is taken over all measurable simple functions s such that 0 \leq s \leq f.

Let f be measurable, and consider the two integrals

(55) $$\int_E f^+ d\mu \mbox{ and} \int_E f^- d\mu$$

where $f^+ = max(f,0)$ and $f^-= min(f,0)$. If at least one of the two integrals is finite, we define

(56) $$\int_E f d\mu = \int_E f^+ d\mu - \int_E f^- d\mu$$

If both integrals on the RHS are finite, we say the f is Lebesgue integrable.

## The Attempt at a Solution

Apparently this should follow directly from the definition, although I am having trouble figuring out why. Assume B is a bound for f. Then all of the c_i have to less then or equal to B (at least if E_i is nonempty). Then we someone need to get a bound for the I_E. So, we probably want to invoke the countable additivity of the set function mu but we then need to know that all of the $E_i \cap E$ are disjoint and we do not know that, right?

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## Answers and Replies

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Homework Helper
Try to invoke the monotonicity of the integral.

The definition of a simple function is that the Ei are disjoint. If they're not, you can split them into new Ei that are disjoint. For instance, if
E1 = (0,1/2), c1 = 1
E2 = (1/4, 3/4), c2 = 2
E3 = (1/2, 1), c3 = 1.
you can easily separate the Ei and ci so that the new Ei are disjoint.

Also, the reason that this follows directly from the definition is that it is the definition. What are you trying to prove?