Definition

Let $\mathcal{B}$ be the σ-algebra of subsets of $\mathbb{R}$. For $a\in\mathbb{R}$, define $\delta_{a}:\mathcal{B}\to[0,\infty)$ by $$\delta_{a}(E)=\begin{cases} 1&\text{if }a\in E \\ 0 & \text{if }a\notin E \end{cases}$$then $\delta_{a}$ is a measure and is called the Dirac measure at $a$.