interchange integral and inf

Question

Let $\mathbb{S}^{n-1}$ be the unit sphere on $\mathbb{R}^n$, A be a positive define diagonal $n\times n$ matrix, and $\mu$ is a probabilistic measure on $\mathbb{S}^{n-1}$. So is it correct to interchange the integral and inf on the following expression: $$\int_{\mathbb{S}^n\ \ } \inf_{y\in\mathbb{S}^{n-1}}\Big( y^TAy-2\langle x,y\rangle \Big)d\mu(x)$$

Answer 1

No. Integral of an infimum is less than or equal to the infimum of the integrals but equality holds only in very special cases.