I am reading an introduction to random matrices.
In the definition of Lebesgue measure of a Hermitian matrix
$$d M = \prod_{1\leq i < j\leq n} d(\Re M_{ij}) d(\Im M_{ij})\prod_{i=1}^n dM_{ii}$$
we have the product of measures of the real and complex part of an entry. Why isn't it the addition?
My understanding is that considering a random matrix, the measure of each entry (an iid variable) is a probabilistic measure indicating a probabilistic distribution, and for an complex entry we assume that the real and complex parts are independently random. Since $P(A\cap B)=P(A)P(B)$, we have the above definition.