Probability Space

A probability space is the mathematical structure used to define a model of uncertainty in probability theory. It is a triple $(\Omega,\mathcal F,P)$, where $\Omega$ is a sample space, $\mathcal F$ is a sigma-algebra of events, and $P$ is a probability measure. These components specify which outcomes belong to the model, which collections of outcomes have defined probabilities, and how those probabilities are assigned. The framework connects probability with measure theory, accommodating discrete outcomes, continuous quantities, and infinite sequences within one formalism. (ocw.mit.edu)

Components and axioms

The sample space $\Omega$ is a nonempty set whose elements $\omega$ represent elementary outcomes. An outcome may be a die face, a sequence of coin tosses, or an entire trajectory of a random system. An event is a subset $A\in\mathcal F$; it occurs when the realized outcome belongs to $A$. Outcomes and events are therefore distinct: an outcome is an element, whereas an event is a measurable set of elements. (ocw.mit.edu)

The event collection $\mathcal F$ contains $\Omega$, is closed under complements, and is closed under countable unions. These requirements also imply closure under countable intersections and inclusion of the empty set. The pair $(\Omega,\mathcal F)$, without a probability measure, is called a measurable space. (math.mit.edu)

A probability measure is a function $P:\mathcal F\to[0,1]$ satisfying normalization, $P(\Omega)=1$, and countable additivity: for any pairwise disjoint events $A_1,A_2,\ldots$,

These axioms imply

and monotonicity: $A\subseteq B$ implies $P(A)\leq P(B)$. (math.mit.edu)

Countable additivity also governs limits of events. If $A_n$ increases to $A=\bigcup_nA_n$, then $P(A_n)\to P(A)$. If $A_n$ decreases to $A=\bigcap_nA_n$, the same convergence holds. These continuity properties distinguish countably additive probability from frameworks requiring only finite additivity. (math.mit.edu)

Discrete and continuous examples

For a fair six-sided die,

Here $2^\Omega$ is the power set, containing every subset of $\Omega$. The event “an even result” is $\{2,4,6\}$, with probability $1/2$. Fairness is encoded in $P$, not in the outcome set itself. A biased die can use the same $\Omega$ and $\mathcal F$ with different probabilities. (ocw.mit.edu)

More generally, on a finite or countable sample space, nonnegative weights $p_\omega$ summing to one define

This is the structure underlying a discrete probability mass function. (ocw.mit.edu)

For a uniform selection from $[0,1]$, one can take $\Omega=[0,1]$, let $\mathcal F$ be its Borel sigma-algebra, and use Lebesgue measure restricted to the interval. Then $P([a,b])=b-a$ for $0\leq a\leq b\leq1$. Each singleton has probability zero, although the whole interval has probability one. There is no contradiction: countable additivity does not require summation over an uncountable family of singletons. (ocw.mit.edu)

The sigma-algebra matters in continuous models. Under the usual set-theoretic assumptions, assigning length consistently to every subset of an interval while retaining countable additivity and translation invariance is impossible. Restricting probabilities to measurable sets avoids this obstruction. Borel sets form the smallest sigma-algebra containing all open sets. (math.mit.edu)

Random variables and distributions

A real-valued random variable on a probability space is a measurable function $X:\Omega\to\mathbb R$. Measurability means that $X^{-1}(B)\in\mathcal F$ for every Borel set $B$. Consequently, statements about the observed value, such as $X\leq x$, correspond to events with defined probabilities. (ocw.mit.edu)

The distribution of $X$ is the induced measure

Its cumulative distribution function is $F_X(x)=P(X\leq x)$. A distribution describes the probabilities of a variable’s values; the underlying space additionally supports relationships between variables. Two variables with identical individual distributions need not have the same joint distribution. (ocw.mit.edu)

The expected value is defined by the Lebesgue integral,

when this integral is well-defined; a finite expectation requires integrability. Thus, sums for discrete variables and integrals for continuous variables are instances of the same measure-theoretic operation. (ocw.mit.edu)

Independence and conditioning

Events $A$ and $B$ exhibit statistical independence when

Independence is a property of the measure and events, not merely of the sample space. It differs from disjointness: disjoint events of positive probability cannot be independent. (ocw.mit.edu)

For $P(B)>0$, conditional probability is

As a function of $A$, this defines another probability measure on $\mathcal F$. When $P(B)=0$, the ratio is undefined; conditioning on variables or information instead requires more general constructions, such as conditional expectation. (ocw.mit.edu)

Null events and completeness

A property holds almost surely when the event on which it fails has probability zero. This does not mean that it holds at every outcome: a nonempty set may have zero probability. A probability space is complete if every subset of every measurable zero-probability set is also measurable. Any probability space can be completed by adding these subsets and assigning them probability zero, without changing the probabilities of its original events. (ocw.mit.edu)