**Setup**
----------
Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by 
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})
,
$$
for a fixed convex and lower-semicontinuous function $g$
.  Here $\mu$ is a (if acceptable $\sigma$-)finite measure on $\mathcal{X}$.  

**Question**
----------

Under what conditions can the interchanging of the 
$$
\inf_{f \in U}\int_{x \in \mathcal{X}}f^2(x)\mu(dx)
 = 
\int_{x \in \mathcal{X}}\inf_{f \in U}f^2(x)\mu(dx).
$$
In other words....when can the infimum be taken pointwise?