Dirac and counting measure

Question

I want to make more practice about how can find the integral with respect to counting measure and Dirac measure.

Any suggestion will be appreciated

Answer 1

Let us start with the Dirac measure. Let $\delta_a(A)=0$ if $a\not\in A$ and $\delta_a(A)=1$ if $a\in A$. Then $$\int_X f\text{d}\delta_a=\int_{X\setminus\{a\}}f\text{d}\delta_a+\int_{\{a\}}f\text{d}\delta_a=0+\int_{\{a\}}f(a)\text{d}\delta_a=f(a)\int_{\{a\}}\text{d}\delta_a=f(a)$$

Now try to compute the integral wrt. the counting measure. Hint: $$\int_X f\text{d}(\mu+\nu)=\int_X f\text{d}\mu+\int_X f\text{d}\nu.$$