= Outer expectation
{title2=$\mathbb E^*Z=\inf\{\mathbb EY:Y\geq Z,\ Y\text{ measurable}\}$}
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 $\mathbb E_*Z=-\mathbb E^*(-Z)$. 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.
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