Does anyone know a proof of the following measure theory fact?
Let $\mu$ be a finite measure on $\mathbb{R}^n$ and $T_i:\mathbb{R}^n\to\mathbb{R}^n$, $i=1,2,3$, measurable functions. If \begin{equation}\frac{1}{2}\left(\int_{\mathbb{R}^n}\varphi(x,T_1(x))\,d\mu +\int_{\mathbb{R}^n}\varphi(x,T_2(x))\,d\mu \right) = \int_{\mathbb{R}^n}\varphi(x,T_3(x))\,d\mu\end{equation} for all $\varphi \in C_c(\mathbb{R}^n\times \mathbb{R}^n)$, them $T_1(x) = T_2(x) = T_3(x)$, for $\mu-a.e.$