Integral of a simple function (measure theory)

From Maths
Jump to: navigation, search
(Unknown grade)
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
DESPERATELY needs fleshing out


For a simple function in its standard representation, say [ilmath]f:=\sum^n_{i=0}x_i\mathbf{1}_{A_i}[/ilmath] then the [ilmath]\mu[/ilmath]-integral, [ilmath]I_\mu:\mathcal{E}^+\rightarrow\mathbb{R} [/ilmath] is[1]:

  • [math]I_\mu(f):=\sum^n_{i=1}x_i\mu(A_i)\in[0,\infty][/math]

Note that this is independent of the particular standard representation of [ilmath]f[/ilmath].

Proof of claims

Claim 1: the [ilmath]\mu[/ilmath]-integral of a simple function is independent of which standard representation it is evaluated as.

See next

See also


  1. Measures, Integrals and Martingales - René L. Schilling