Reference of delta function as a measure

Question

Any one know the reference of direct delta function as a measure, I need a graduate math rigorous version not physical one.

Answer 1

Do you mean the Dirac-measure? $$ \delta_{x_0}(A) = \begin{cases} 1 & \text{, if }x_0 \in A \\ 0 & \text{, if }x_0 \notin A \end{cases} $$ Obviously, $\Omega$ is some set, $x_0 \in \Omega$ and $A\in \Sigma$ for some $\sigma$-algebra $\Sigma \subseteq 2^\Omega$.

If you use $\delta_{x_0}$ to integrate, it works like the delta distribution: $$ \int_{\Omega} f~\mathrm{d}\delta_{x_0} = f(x_0) $$ for some measurable $f:\Omega \rightarrow \mathbb{R}$.

This is because of $\delta_{x_0} (\Omega \setminus \lbrace x_0 \rbrace) = 0$ and $$ \int_{\Omega} f~\mathrm{d}\delta_{x_0} = \underbrace{\int_{\Omega \setminus \lbrace x_0 \rbrace} f~\mathrm{d}\delta_{x_0}}_{=0} + \int_{\lbrace x_0 \rbrace} f~\mathrm{d}\delta_{x_0} = f(x_0)\delta_{x_0}(\lbrace x_0 \rbrace) = f(x_0) $$ by definition of the integral.