Proof of unitairity of regular representation

Question

Let $G$ be a compact group and $\varphi:G\to L^2(G), \ \sigma\mapsto L_\sigma$ the regular representation.

In the proof of unitairity of $\varphi$ i dont understand one step.

First, the haar measure $\mu$ of $G$ satisfies $$\int_G L_\sigma f d\mu=\int_G f d\mu$$ for all $f$ and $\sigma \in G$.

For the unitairity we have the following $$\langle L_x f,L_xg\rangle=\int_G L_x f(y)\overline{L_xg(y)} dy=\int_G f(x^{-1}y)\overline{g(x^{-1}y)}dy=\int_G f(x)\overline{g(y)}dy=\langle f,g\rangle.$$

why is there equality $\int_G f(x^{-1}y)\overline{g(x^{-1}y)}dy=\int_G f(y)\overline{g(y)}dy$ ?

How to use invariance of the haar measure of $G$?

Thanks

Answer 1

What you are asking is a consequence of the following property: If $f:G\to \mathbb{C}$ is integrable, then $$ \int_G f(yx)\; dx = \int_G f(x)\; dx \quad \text{for all } y\in G, $$ which in turn follows from the left invariance of the Haar measure. To prove this we first consider $f$ to be a simple function, namely $f=1_A$ for some Borel set $A$, and note that $$ \int_G 1_A(yx)\; dx = \int_G 1_{y^{-1}A}(x)\; dx = \mu(y^{-1}A) = \mu(A) = \int_G 1_A(x) \; dx. $$ Then the result is true for linear combinations of such functions and by monotone convergence, for positive measurable functions and finally for any integrable function.