Integration with respect to Dirac measure using a step function I have troubles to prove that $\int f(x) d\delta_x(A)=f(x)$ using a step function like $g(x)=\sum_i c_i \mathbb{1_{A_i}}(x) $.
I have tried this :
$$\int g(x)d\delta_x(A)=\int \sum_i c_i \mathbb{1_{A_i}}(x) d \delta_x(A)=\sum_i c_i\int  \mathbb{1_{A_i}}(x) d \delta_x(A)=\sum_i c_i\int_ \mathbb{{A_i}}d \delta_x(A)$$
And as we know in general that $\int_A dP(x)= P(A)$ we get something like this (This is where I get confused):
$$\int g(x)d\delta_x(A)=\sum_i c_i\delta_ \mathbb{{A_i}}(A)$$
which doesn't lead to the result $\int g(x)d\delta_x(A)=g(x)$
Any comments? Help? Thank you

You want to show $\int_X f d \delta_x=f(x)$. First consider a simple function $s=\sum_i c_i 1_{A_i}$ where the $A_i$ are disjoint. For convenience we can assume WLOG that $\bigcup A_i=X$ (by taking a $c_i$ to be zero if need be). Then there is exactly one $i$ with $x \in A_i$, call this $i(x)$. Then $\int_X s d \delta_x = c_{i(x)} \delta_x(A_{i(x)})=c_{i(x)}=s(x)$. Now extend in the usual way.
