Consider the unitsquare in the Euclidean planeR2, S = × . Consider the probability measure μ defined on S by the restriction of two-dimensional Lebesgue measure λ2 to S. That is, the probability of an event E ⊆ S is simply the area of E. We assume E is a measurable subset of S. Consider a one-dimensional subset of S such as the line segmentLx = × . Lx has μ-measure zero; every subset of Lx is a μ-null set; since the Lebesgue measure space is a complete measure space, While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" Lx is the one-dimensional Lebesgue measure λ1, rather than the zero measure. The probability of a "two-dimensional" event E could then be obtained as an integral of the one-dimensional probabilities of the vertical "slices" E ∩ Lx: more formally, if μx denotes one-dimensional Lebesgue measure on Lx, then for any "nice" E ⊆ S. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.
Statement of the theorem
will denote the collection of Borel probability measures on a metric space The assumptions of the theorem are as follows:
Let π : Y → X be a Borel-measurable function. Here one should think of π as a function to "disintegrate" Y, in the sense of partitioning Y into. For example, for the motivating example above, one can define, which gives that, a slice we want to capture.
Let ∈ P be the pushforward measure = π∗ = μ ∘ π−1. This measure provides the distribution of x.
The conclusion of the theorem: There exists a -almost everywhere uniquely determined family of probability measures x∈X ⊆ P, which provides a "disintegration" of into ), such that:
the function is Borel measurable, in the sense that is a Borel-measurable function for each Borel-measurable set B ⊆ Y;
μx "lives on" the fiber π−1: for -almost all x ∈ X,
The original example was a special case of the problem of product spaces, to which the disintegration theorem applies. When Y is written as a Cartesian productY = X1 × X2 and πi : Y → Xi is the natural projection, then each fibre π1−1 can be canonicallyidentified with X2 and there exists a Borel family of probability measures in P∗ such that which is in particular and The relation to conditional expectation is given by the identities
Vector calculus
The disintegration theorem can also be seen as justifying the use of a "restricted" measure in vector calculus. For instance, in Stokes' theorem as applied to a vector fieldflowing through a compact surface Σ ⊂ R3, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ3 on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ3 on ∂Σ.
The disintegration theorem can be applied to give a rigorous treatment of conditional probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.
