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Let $F$ be a cone of upper bounded upper semicontinuous functions on a compact set set $X$ containing all the constants. Let $z\in X $ and define a class of positive measure by $$M_z^F=\{ \mu : u(z)\leq \int_X u d\mu \mbox{ for all } u\in F\} $$

Let $g: X\rightarrow R$ and set $$Sg(z)=\sup\{u\in F, u\leq g\}$$ $$ Ig(z)=\inf\{ \int_X g d\mu , \mu\in M^F_z\} \}$$

The following statement correspond to Edwards theorem : With $F$ as above , and if $g $ is a bounded Borel function on $X$, then $Sg(z)\leq Ig(z)$. If $g$ is lower semicontinuous , then $Sg=Ig$.

One can find an example in Frank Wikstrom Paper (Jensen measure and boundary value of plurisubharmonic functions) and also in Ransford paper showing that if $g$ is upper semi-continuous then $Sg\not= Ig$. \

Can any body tell me whether the upper semi-continuous regularizations coincide in general case? i.e $(Sg)^*(z)=(Ig)^*(z)$ (or when g is upper semi-continuous).

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