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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.

Charlie Chang
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    When an author gives me a definition I usually don't ask myself immediatley the question why he or she has not done it differently. The above expression reminds me of the product measure of $n(n+1)/2+n(n-1)/2$ Lebesgue measures. Looks OK. – Kurt G. May 14 '22 at 18:54
  • I tend to be puzzled by the motivations behind definitions, because if I don't know them the related theorems do not easily make sense to me. When would you usually ask yourself such questions? $\quad$ For me, I usually try to give an (even inaccurate) understanding of motivations, and then proceed and update my understanding, similar to a Bayesian approach. – Charlie Chang May 14 '22 at 18:59
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    At the earliest when it comes to the theorems. Keep reading and see what comes. – Kurt G. May 14 '22 at 19:03
  • There seem to be sound reasons for those definitions in analysis and part of topology, but they are manifest only much later (sometimes in the next course, e.g. functional analysis is possibly not well contextualized until analysis of PDE..) – Charlie Chang May 15 '22 at 04:50

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