In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures. In a sense, "disintegration" is the opposite process to the construction of a product measure.

Consider the unit square ${\displaystyle S=[0,1]\times [0,1]}$ in the Euclidean plane ${\displaystyle \mathbb {R} ^{2}}$. Consider the probability measure ${\displaystyle \mu }$ defined on ${\displaystyle S}$ by the restriction of two-dimensional Lebesgue measure ${\displaystyle \lambda ^{2}}$ to ${\displaystyle S}$. That is, the probability of an event ${\displaystyle E\subseteq S}$ is simply the area of ${\displaystyle E}$. We assume ${\displaystyle E}$ is a measurable subset of ${\displaystyle S}$.

Consider a one-dimensional subset of ${\displaystyle S}$ such as the line segment ${\displaystyle L_{x}=\{x\}\times [0,1]}$. ${\displaystyle L_{x}}$ has ${\displaystyle \mu }$-measure zero; every subset of ${\displaystyle L_{x}}$ is a ${\displaystyle \mu }$-null set; since the Lebesgue measure space is a complete measure space, $${\displaystyle E\subseteq L_{x}\implies \mu (E)=0.}$$

While true, this is somewhat unsatisfying. It would be nice to say that ${\displaystyle \mu }$ "restricted to" ${\displaystyle L_{x}}$ is the one-dimensional Lebesgue measure ${\displaystyle \lambda ^{1}}$, rather than the zero measure. The probability of a "two-dimensional" event ${\displaystyle E}$ could then be obtained as an integral of the one-dimensional probabilities of the vertical "slices" ${\displaystyle E\cap L_{x}}$: more formally, if ${\displaystyle \mu _{x}}$ denotes one-dimensional Lebesgue measure on ${\displaystyle L_{x}}$, then $${\displaystyle \mu (E)=\int _{[0,1]}\mu _{x}(E\cap L_{x})\,\mathrm {d} x}$$ for any "nice" ${\displaystyle E\subseteq S}$. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

(Hereafter, ${\displaystyle {\mathcal {P}}(X)}$ will denote the collection of Borel probability measures on a topological space ${\displaystyle (X,T)}$.) The assumptions of the theorem are as follows:

The conclusion of the theorem: There exists a ${\displaystyle \nu }$-almost everywhere uniquely determined family of probability measures ${\displaystyle \{\mu _{x}\}_{x\in X}\subseteq {\mathcal {P}}(Y)}$, which provides a "disintegration" of ${\displaystyle \mu }$ into ${\displaystyle \{\mu _{x}\}_{x\in X}}$, such that:

The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.

When ${\displaystyle Y}$ is written as a Cartesian product ${\displaystyle Y=X_{1}\times X_{2}}$ and ${\displaystyle \pi _{i}:Y\to X_{i}}$ is the natural projection, then each fibre ${\displaystyle \pi _{1}^{-1}(x_{1})}$ can be canonically identified with ${\displaystyle X_{2}}$ and there exists a Borel family of probability measures ${\displaystyle \{\mu _{x_{1}}\}_{x_{1}\in X_{1}}}$ in ${\displaystyle {\mathcal {P}}(X_{2})}$ (which is ${\displaystyle (\pi _{1})_{*}(\mu )}$-almost everywhere uniquely determined) such that $${\displaystyle \mu =\int _{X_{1}}\mu _{x_{1}}\,\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right)=\int _{X_{1}}\mu _{x_{1}}\,\mathrm {d} (\pi _{1})_{*}(\mu )(x_{1}),}$$ which is in particular^{[clarification needed]} $${\displaystyle \int _{X_{1}\times X_{2}}f(x_{1},x_{2})\,\mu (\mathrm {d} x_{1},\mathrm {d} x_{2})=\int _{X_{1}}\left(\int _{X_{2}}f(x_{1},x_{2})\mu (\mathrm {d} x_{2}\mid x_{1})\right)\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right)}$$ and $${\displaystyle \mu (A\times B)=\int _{A}\mu \left(B\mid x_{1}\right)\,\mu \left(\pi _{1}^{-1}(\mathrm {d} x_{1})\right).}$$