= Solution
Work on a <filtered probability space> $(\Omega,\mathcal F,(\mathcal F_n)_{n\geq0},\mathbb P)$. A <submartingale> is a real-valued <stochastic process> satisfying the following three conditions: $Y_n$ is $\mathcal F_n$-measurable, $\mathbb E|Y_n|<\infty$, and
$$
\boxed{\mathbb E[Y_{n+1}\mid\mathcal F_n]\geq Y_n\quad\text{almost surely}.}
$$
Thus it is <adapted>, has <integrable random variables> at every time, and its conditional expected next value is at least its current value. Repeated conditioning gives $\mathbb E[Y_m\mid\mathcal F_n]\geq Y_n$ for every $m\geq n$. \b[The inequality concerns <conditional expectation>; individual <sample paths> need not increase.] If no <filtration> is specified, the definition can be applied to the <natural filtration> generated by the process.
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