The Dirac measure is defined by $$\delta_x(A)= \begin{cases} 1 &\text{if $x \in A$}\\ 0 &\text{if $x \notin A$}\\ \end{cases}$$ Let $f:X\rightarrow \mathbb{R}$ be a function. Can anyone show me why $\int f \, d\delta_x=f(x)$? Any help is appreciated.
[Math] Integration with respect to Dirac measure
measure-theory
Best Answer
Hint If $g=\sum_i c_i 1_{A_i}$ is a step function, then
$$\int g d \delta_x = \sum_i c_i \delta_x(A_i)=\sum_i c_i 1_{A_i}(x)=g(x) $$