7.11 Let $P(A)=\int_{-\infty}^\infty1_A(x)f(x)\,dx$ for a nonnegative function $f$ with $\int_{-\infty}^\infty f(x)\,dx=1$. Let $A=\{x_0\}$, a singleton (that is, the set $A$ consists of one single point on the real line). Show that $A$ is a Borel set and also a null set (that is, $P(A)=0$).
Could anyone help me solve it? Thank you!
$A = \{x_0\}$ can be constructed using $\bigcap_{n=1}^\infty (x_0 - \frac{1}{n}, x_0 + \frac{1}{n})$, so it has to be a Borel set. Since $1_A(x)f(x)$ is zero almost everywhere, $P(A) = 0$.