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Re: Taking math courses just for fun?
If you can handle your research while taking (and doing well in) these difficult courses, and you have everything else ready (including letters, which will be very important for you), why not? If you want to be a hardcore theorist or theoretical econometrician, there may be some benefit to functional analysis, measure theoretic probability, and convex analysis/optimization. If you want to do applied work, there are likely better courses for you to take.