Is it true: $\min_{y}E_Xg(X,y) \le \max_{x}E_Yg(x, Y)$?

Question

In my research I stack with this problem.

Let $X, Y$ are random values, and $g(x, y)$ some function (with values in $\mathbb{R}$). Is this inequality true?

$$\inf_{y_0}E_Xg(X, y_0)\le \sup_{x_0}E_Yg(x_0, Y)$$