I have come across the following definition of martingale in various texts:
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $T$ be a fixed positive number, and let $\mathcal{F}(t)$, $0\leq t\leq T$, be a filtration of sub-$\sigma$-algebras of $\mathcal{F}$. An adapted stochastic process $M(t),0\leq t\leq T$, is said to be a martingale if
$1.$ $\mathbb{E}(|M(t)|)<\infty$, for all $t\geq0$
$2.$ $\mathbb{E}(M(t)|\mathcal{F}(s))=M(s)$ almost surely, for all $0\leq s\leq t\leq T$.
What does the second condition mean in the measure-theoretic sense?
Does this mean $\mathbb{P}\big(\{w\in\Omega:\mathbb{E}(M(t)|\mathcal{F}(s))(w)=M(s)(w),~\text{for all}~0\leq s\leq t\leq T\}\big)=1?$
or
we only have: For any fixed $s,t$ with $0\leq s\leq t\leq T$, $\mathbb{P}\big(\{w\in\Omega:\mathbb{E}(M(t)|\mathcal{F}(s))(w)=M(s)(w)\}\big)=1?$