Integral of a simple function (measure theory)/Definition

From Maths
< Integral of a simple function (measure theory)
Revision as of 17:05, 17 March 2016 by Alec (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
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:
Quickly written. Needs to go something like "using every simple function has a standard representation" and then showing that the integral is the same for each standard representation is what needs to be done

Definition

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].

References

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