Lebesgue decomposition
Topic of Random variables and laws, part of Stochastic processes
Lebesgue decomposition
| Type | Topic |
| Taught in | Random variables and laws |
| Part | Stochastic processes |
Every distribution on the real line is uniquely a mixture of a discrete part (atoms), an absolutely continuous part (with a density) and a singular continuous part. The distribution function correspondingly has jumps, a differentiable part, and a part that increases on a set of length zero.
In Random variables and laws
If X has a density f, then F_X(x)=\int_{-\infty}^x f(u)\,du, and F_X'(x)=f(x) at points where the derivative exists. A density is allowed to exceed 1. Its integral over the line equals 1. A law can also mix atoms with a density. The distribution function then has jumps and continuously increasing pieces. The Lebesgue decomposition splits a general law into a discrete part, an absolutely continuous part, and a singular continuous part.
Used in
See also
Cumulative distribution function · Measurable random variable · Simple variables and expectation · Kolmogorov extension theorem · Atom of a distribution