multi-dimensional impulse with kernel

Question

We know that a 1d dirac function "extracts" the function of an integral, but what about for nd?

$$\int \cdots \int f(x_1, \cdots, x_n) \prod_{i=1}^{n} \delta(x - x_i) dx_i$$

is this equal to $f(x,\cdots,x)$?

Answer 1

The delta measure works identically in any number of dimensions.