# Homework Help: Integral using Lebesgue Measure

1. Jun 26, 2010

### lmedin02

1. The problem statement, all variables and given/known data
Find the integral of the function f(x)=3 when x is rational and 2 when x is irrational on the interval [0,1].

2. Relevant equations

3. The attempt at a solution
So I partition [0,1] into two disjoint sets A and B. $$A=[0,1] \cap Q$$ and $$B=[0,1] \cap Q^{c}$$.
Now the integral should be equal to 3 length(A) + 2 length(B). Since A is a countable set then its measure is zero. But I do not understand how to calculate the measure of B since it is uncountable.

2. Jun 26, 2010

### Count Iblis

But you know what the measure of [0,1] is...

3. Jun 26, 2010

### lmedin02

Thanks for the hint. So 1=len([0,1])=len(A)+len(B)=len(B).