How is the Dirac function different from the indicator function My question is straightforward though I cant find an answer online. What is the difference between an indicator function and a Dirac function?

Let $A=\{0\}$ and indicator function on $A$ be $I_A$. Measure theoretically, $I_A=0$ a.e (almost everywhere) and
$\int_{\mathbb{R}}I_A=0.$
Dirac function is point mass at origin (in fact, it is a functional on set of compactly supported smooth functions) and can not be defined point-wise. Most importantly $\int_{\mathbb{R}}\delta(x)dx=1$ .  
