The definition of Density of States

Question

The density of states (DOS) is generally defined as $D(E)=\frac{d\Omega(E)}{dE}$, where $\Omega(E)$ is the number of states. But why DOS can also be defined using delta function, as $$D(E)~=~\sum\limits_{n} \int \frac{d^3k}{(2\pi)^3}\delta(E-\epsilon_n(\mathbf{k}))?$$

Answer 1

OP's equality involving a delta function is probably easier to appreciate in its equivalent integrated form

$$\int \!d\Omega(E) ~f(E)~=~ \int \!dE~D(E) ~f(E)~ ~=~\sum_{n} \int \frac{d^3k}{(2\pi)^3}f(\epsilon_n(\mathbf{k})), $$ where $f(E)$ is an arbitrary function.