When writing an article I encounter what is essentially a conditional expectation - function defined on a bounded interval (not necessarily of unit length) with Lebesgue measure, but information about it is restricted to sets of a given sub-sigma-algebra, which is usually not finite. The article does not otherwise use probabilistic terminology, so it would be nice to not make it necessary for readers to be familiar with conditional expectations.

**Does there exist established terminology or notation for the conditional expectation outside probabilistic context?**

I will certainly mention the probabilistic interpretation. Otherwise, the paper uses classical epsilon-delta analysis, functional analysis, and a smattering of calculus of variations. The motivation is in PDE and inverse problems.

The function turns out to be in all $L^p$ spaces, and in particular in $L^\infty$.