Probability space ($(\Omega ,\mathcal{F},\mathbb{P})$)

A probability space is a measure space whose total measure is one. Its measurable subsets are events, and its measure $\mathbb{P}$ is their probability.

An almost sure event is a measurable event of probability one. Countable intersections of such events still have probability one, allowing a conclusion to hold simultaneously for countably many parameter choices. Further deterministic reasoning may then extend it to uncountably many parameters, as in Brownian Hölder regularity or nowhere monotonicity of Brownian motion. An uncountable intersection need not preserve probability one without such additional reasoning.

A probability measure is a measure with total mass one.

A probability measure is tight if compact sets carry arbitrarily nearly all its mass. Every Borel probability measure on a Polish space is tight. The stronger family condition is uniform tightness.

For càdlàg processes on a finite horizon, compact containment together with uniformly small increments over pairs of bounded stopping times whose separation tends to zero is a sufficient criterion for tightness in the Skorokhod J1 topology. The increment condition is in probability, not merely at deterministic times. If in addition the largest jump tends to zero in probability, every limiting law is supported on continuous paths.

The same compact set must work simultaneously for every measure in the family. Prokhorov's theorem relates this condition to relative compactness for the weak topology of probability measures on a Polish space.

A zero-probability event has probability measure zero. It need not be empty: for a continuously distributed random variable, equality with a specified value is such an event. The complementary event occurs almost surely.

For observations or orbit points $x_0,\dots ,x_{N-1}$, the empirical measure is ${\nu }_N=N^{-1}{\sum }_{j=0}^{N-1}{\delta }_{x_j}$. Integrating a test function gives its sample average. For orbit points $x_j=T^{j}x$, the difference between $T_{\ast }{\nu }_N$ and ${\nu }_N$ telescopes to $({\delta }_{T^{N}x}-{\delta }_x)/N$, so on a compact metric space with continuous $T$, every weak limit is invariant.

If finite Borel measures $\mu$ and $\nu$ on $\mathbb{R}$ satisfyfor every compactly supported continuous function $f$, then $\mu =\nu$. This follows from the uniqueness part of the Riesz representation theorem; alternatively, continuous functions that approximate indicators from above and below recover the measure of every interval and hence every Borel set.

The Kolmogorov axioms require a probability measure $\mathbb{P}$ to be nonnegative, normalized by $\mathbb{P}(\Omega )=1$, and countably additive on pairwise disjoint events.

For a countable measurable partition $(B_i)$ with positive probabilities,

An event is a measurable subset of a probability space.

The event that $A$ does not occur has probability $1-P(A)$. Conditioning on an event positively associated with $A$ negatively associates its complement with $A$.

Two events are disjoint when they cannot both occur. For a finite family of pairwise disjoint events, the probability of their union is the sum of their probabilities, so that sum is at most one. This condition differs from the condition defining independent events.

On a product space ${\mathbb{R}}^T$, the cylinder sigma-algebra is generated by coordinate projections. Every event in it depends on at most countably many coordinates.