 Solution

ID: past-exam-of-the-real-and-harmonic-analysis-course-of-the-university-of-oxford/2008/3/f/solution

Convolution preserves probability measures: its defining pushforward is nonnegative and its total mass is the product of the two unit masses. For every fixed integer

n

the previous coefficient identity gives

μ_{k} * ν_{k} (n) = μ_{k} (n) ν_{k} (n) ⟶ μ (n) ν (n) = μ * ν (n) .

The probability version of the Fourier coefficient criterion therefore proves

μ_{k} * ν_{k} ⇀ * μ * ν .

The fixed compact underlying group and the uniform probability masses supply precisely the bounds needed in the dense-test argument.

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