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There are a lot of places explaining Haar Distribution but they require some basics of Borel fields, Lie fields, etc. I am not a math major, so I do not have exposure to these concepts.

Here is my understanding of Haar distribution:

Take a $N×N$ matrix, say $M$, of i.i.d. standard Gaussian random variables.One can take a QR decomposition of $M$ and get an orthogonal Matrix $Q$. People claim that the matrix $Q$ is a Haar measure over O(N).

A key property: $Q$is left-invariant, meaning, for any $R∈O(N)$, $Q$and $RQ$ have the same distribution.

What is the proof of this invariance property? ( in terms of matrix algebra, if possible)

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