Lebesgue Decomposition Theorem

Definition

A Distribution $F$ can be decomposed as the following $F=F_{ac}+F_{pp}+F_{sing}$ Where:

• $F_{ac}$: absolutely continuous part (absolutely continuous w.r.t $\lambda$)
• $F_{pp}$: pure point part: (absolutely continuous w.r.t ${\#}_X$)
• $F_{sing}$: singular continuous part (e.g. cantor distribution)

Facts

$F_{pp}$ has countable discontinuous points

If a Distribution Function $F=F_{ac}$, then there exists corresponding continuous Random Variable $X$ and PDF

If a Distribution Function $F=F_{pp}$, then there exists corresponding discrete Random Variable $X$ and generalized PDF or PMF

If a Distribution Function $F=F_{ac}+F_{pp}$, then there exists corresponding mixed Random Variable $X$ and generalized PDF