Conditional expectation may be computed on each part of a partition and then averaged. The example is deliberately concrete; it is a test of the statement, not a substitute for it.
Start locally
A probability space \((\Omega,\mathcal F,\mathbf P)\) separates outcomes, measurable events, and their probabilities. A random variable \(X:\Omega\to\mathbf R\) must be measurable.
I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.
Compute before generalising
The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.
The global view
The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.
Edge conditions
Conditioning on a probability-zero event cannot be done by naïvely dividing. Conditional densities and regular conditional probabilities require additional structure.
A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.