Conditional probability is probability after observing information. Regular conditional distributions make this idea a measurable kernel rather than only an almost-sure identity.
Conditioning on a sigma-field
Conditional probabilities are conditional expectations of indicators.
Definition. For an event \(A\in\mathcal F\) and a sub-\(\sigma\)-field \(\mathcal G\), define \[ P(A\mid\mathcal G)=\mathbb E[1_A\mid\mathcal G]. \] It is a \(\mathcal G\)-measurable random variable, unique almost surely.
Regular conditional laws
A regular conditional distribution chooses versions consistently over all target sets.
Theorem. If \(X\) takes values in a standard Borel space \(S\), then there exists a probability kernel \(K(\omega,B)\) such that \(K(\cdot,B)\) is \(\mathcal G\)-measurable, \(K(\omega,\cdot)\) is a probability measure, and \(K(\cdot,B)=P(X\in B\mid\mathcal G)\) a.s. for every \(B\in\mathcal B(S)\).
Proof outline. Reduce through a measurable isomorphism to a Borel subset of \([0,1]\). For every rational \(q\), choose a version \[ F_q(\omega)=P(X\le q\mid\mathcal G)(\omega). \] Because the rationals are countable, one null set can be removed so that \(q\mapsto F_q(\omega)\) is increasing and has the correct limits for every remaining \(\omega\). Define a right-continuous cdf by \[ F_\omega(t)=\inf_{q>t,\ q\in\mathbb Q}F_q(\omega), \] and let \(K(\omega,\cdot)\) be its Lebesgue-Stieltjes measure. Measurability holds first on rational half-lines and extends to all Borel sets by a monotone-class argument. Transport the kernel back through the measurable isomorphism.
Definition. A probability kernel from \((T,\mathcal T)\) to \((S,\mathcal S)\) is a map \(K:T\times\mathcal S\to[0,1]\) such that \(K(t,\cdot)\) is a probability measure for every \(t\), and \(K(\cdot,B)\) is \(\mathcal T\)-measurable for every \(B\).
Theorem (conditioning on a random variable). If \(Y\) is standard Borel valued, then there is a kernel \(K(y,\mathrm{d}x)\), unique \(P_Y\)-almost everywhere, such that \[ P(X\in A,Y\in B) =\int_BK(y,A)\,P_Y(\mathrm{d}y). \] It is denoted \(P_{X\mid Y}(A\mid y)\).
Proof. Apply the regular-conditional theorem with \(\mathcal G=\sigma(Y)\). By the Doob-Dynkin lemma, each \(\sigma(Y)\)-measurable version \(P(X\in A\mid\sigma(Y))\) can be written as \(K(Y,A)\). A countable generator and monotone-class extension choose these versions simultaneously as a kernel. Multiplying by \(\mathbf1_{\{Y\in B\}}\) and taking expectation gives the disintegration formula.
Disintegration formula
Regular conditionals turn conditional expectations into integrals against kernels.
Proposition. If \(K\) is a regular conditional law of \(X\) given \(\mathcal G\), then for nonnegative or integrable \(g\), \[ \mathbb E[g(X)\mid\mathcal G](\omega)=\int_S g(x)\,K(\omega,\mathrm{d}x) \] for a suitable version.
Proof. For indicators \(g=1_B\), this is the defining property of \(K\). Linearity gives simple functions, monotone convergence gives nonnegative measurable functions, and positive-negative decomposition gives integrable functions.
Example (Gaussian conditioning). If \[ \begin{pmatrix}X\\Y\end{pmatrix} \sim N\!\left( \begin{pmatrix}\mu_X\\\mu_Y\end{pmatrix}, \begin{pmatrix} \Sigma_{XX}&\Sigma_{XY}\\ \Sigma_{YX}&\Sigma_{YY} \end{pmatrix} \right) \] and \(\Sigma_{YY}\) is invertible, then \[ X\mid(Y=y)\sim N\!\left( \mu_X+\Sigma_{XY}\Sigma_{YY}^{-1}(y-\mu_Y), \Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX} \right). \] Completing the square in the joint Gaussian density proves the formula. The conditional covariance is the Schur complement and does not depend on the observed value.
Remark. For a single event \(A\), \(P(A\mid\mathcal G)\) always exists as a Radon-Nikodym derivative. A regular conditional law asks for uncountably many eventwise versions to fit together into one countably additive measure for almost every conditioning value. That simultaneous choice can fail on arbitrary measurable spaces; the standard Borel hypothesis is the structural remedy.