Partial sums of independent mean-zero increments have no predictable drift. The point is to make the formal expression readable enough to audit line by line.
Objects and notation
A process \((M_n,\mathcal F_n)\) is a martingale when \(\mathbf E[M_{n+1}\mid\mathcal F_n]=M_n\). Its baseline \(M_0\) models fair evolution relative to the information currently available.
The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).
Push the symbols
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.
Structural reading
The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.
A hypothesis worth keeping
Optional stopping is not valid for every stopping time. Boundedness, integrability, or uniform-integrability conditions prevent hidden mass from escaping at infinity.
This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.