Lebesgue's decomposition theorem

The decomposition theorem of Lebesgue (also Lebesgue decomposition ) shows the disassembly of a - finite measure into an absolutely continuous and singular part with respect to another - finite measure.

Let and be two finite measures on the same measurement space. Then there are two dimensions and so that the cutting

Is uniquely determined. This means that is absolutely continuous with, and means and singular to each other.

The two dimensions and can be calculated using the set of Radon - Nikodym, because after this, there exists a -almost everywhere uniquely determined function. With the amount now are the dimensions: by

Clearly defined.

The decomposition theorem of Lebesgue finds application in the representation theorem in the stochastic.