Applications
Immediately after publishing the Haar theorem, it was used to solve Hilbert's fifth problem which concerns the characterization of Lie groups. Haar measures apply more generally to two fields of mathematics, abstract harmonial analysis and mathematical statistics.
In abstract harmonial analysis, Haar measures are used on locally compact groups, especially with Pontryagin duality. For more on harmonial analysis: https://en.wikipedia.org/wiki/Harmonic_analysis#Abstract_harmonic_analysis For more on Pontryagin duality: https://en.wikipedia.org/wiki/Pontryagin_duality
In statistics, Haar measures are used on prior probabilities for compact groups of transformations. Another area of statistics that Haar measures are used in is conditional inference. For more reading on prior probabilities: https://en.wikipedia.org/wiki/Prior_probability For more reading on conditional inference: https://en.wikipedia.org/wiki/Conditional_inference
For more in depth reading on the various uses of Haar measures: https://en.wikipedia.org/wiki/Haar_measure#Uses
