Measure given by an integral

Question

Find $\mu( \mathbb Q \cap [0,1] )$ and $\mu( \mathbb Q \cup [0,1] )$, where $\mathcal L(\mathbb R)$ is the Lebesgue sigma algebra and the function $\mu: \mathcal L(\mathbb{R})\to [0,\infty]$ maps $A \mapsto \int_{A} x^2 \ d \lambda(x)$.

Just how can I plug the sets in the integral? Thanks

Answer 1

Here are two useful facts:

• If $A$ is a set of measure zero and $f$ is any measurable function, then $ \int_A f \ d\lambda = 0.$

• If $A = [a,b]$ is an interval and $f$ is Riemann integrable, then $\int_A f \ d\lambda$ is equal to the Riemann integral of $f$ over $[a,b]$.

Using the first fact, it should be possible to solve the case $A = \mathbb Q \cap [0,1]$ immediately.

For $A = \mathbb Q \cup [0,1]$, we could split $A$ into two disjoint subsets: $$ A = A_1 \cup A_2,$$ $$ {\rm with \ \ } A_1 = [0,1] {\rm \ \ \ \ and \ \ \ \ } A_2 = \mathbb Q \cap \left( (-\infty, 0) \cup (1, \infty)\right).$$ Now observe that $A_1$ is an interval and $A_2$ is of measure zero...