Dirac delta function and dirac measure I want to know the relationship between these two things. e.g. What's the results of following integrals? (Let $\mu$ Lebesgue measure, $\nu$ Dirac measure)
$$(1)\int\delta_c(x)\mu(dx)$$
$$(2)\int\delta_c(x)\nu(dx)$$
$$(3)\int f(x)\delta_c(x)\mu(dx)$$
$$(4)\int f(x)\delta_c(x)\nu(dx)$$

In analysis and probability, Dirac's delta $\delta_c$ is commonly seen as a function defined on a space of functions. Here are two examples:

*

*In the Theory of distributions for example $\delta_c$ is defined as  $\delta_c:\mathcal{C}^\infty_c(\mathbb{R}^d)\rightarrow\mathbb{R}$ defined as $\delta_c\phi = \phi(c)$.  Clearly $\delta_c$ defines a linear function.


*In measure theory or the thoery of integration, the $\delta_c$ is defined as a measure on a measurable space $(X,\mathscr{B})$ such that $\delta_c(A)=1$ if $c\in A$, $0$ otherwise. It is easily seen that $\delta_c$ induces a linear function on the space of measurable functions on $(X,\mathscr{B})$ be setting $\delta_cf=f(c)$.
your expressions (1)-(4) are not properly defined unless there is some additional context. For instance, if $\delta_c$ and $\mu$ are measures,  it make sense to talk about the product measure $\delta_c\otimes\mu$:
$$
\int f(x,y)\delta_c(dx)\mu(dy)=\int f(c,y)\mu(dy)$$
If  $\delta_c$ is though of as a  distributions, and may talk about $\delta_c*\phi$, for any $\phi\in C^\infty_c(\mathbb{R}^d)$
