Outer expectation (${\mathbb{E}}^{\ast }Z=\inf \{\mathbb{E}Y:Y\ge Z,Y\text{ measurable}\}$)

For a bounded possibly nonmeasurable real function $Z$ on a probability space, its outer expectation is the infimum of expected values of measurable majorants. Its inner expectation is ${\mathbb{E}}_{\ast }Z=-{\mathbb{E}}^{\ast }(-Z)$. They agree with the usual expected value for measurable $Z$. This permits function-space convergence in distribution for an empirical distribution process whose paths are not Borel-measurable in a nonseparable norm topology: both expectations of each bounded continuous test functional must approach its ordinary expectation under the tight measurable limiting law.