It is very common to indicate with $\Gamma_0(A)$ the set of lower semicontinuous convex functions from $A$ to $(-\infty,+\infty]$ with nonempty domain. An example of usage of this notation can be found in "Proximal Splitting Methods in Signal Processing" by Combettes and Pesquet.

What is the role of the subindex $0$ in the short notation?

It is very common to indicate with $\Gamma_0(A)$ the set of lower semicontinuous convex functions from $A$ to $(-\infty,+\infty]$ with nonempty domain. An example of usage of this notation can be found in "Proximal Splitting Methods in Signal Processing" by Combettes and Pesquet.

What is the role of the subscript $0$ in the short notation?