Difference between revisions of "Integral of a simple function (measure theory)/Definition"
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{{Stub page|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}} | {{Stub page|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== | ==Definition== | ||
| − | </noinclude> | + | </noinclude>For a [[simple function (measure theory)|simple function]] in its [[standard representation (measure theory)|standard representation]], say {{M|1=f:=\sum^n_{i=0}x_i\mathbf{1}_{A_i} }} then the {{M|\mu}}-integral, {{M|I_\mu:\mathcal{E}^+\rightarrow\mathbb{R} }} is{{rMIAMRLS}}: |
| − | For a [[simple function (measure theory)|simple function]] in its [[standard representation (measure theory)|standard representation]], say {{M|1=f:=\sum^n_{i=0}x_i\mathbf{1}_{A_i} }} then the {{M|\mu}}-integral, {{M|I_\mu:\mathcal{E}^+\rightarrow\mathbb{R} }} is{{rMIAMRLS}}: | + | |
* {{MM|1=I_\mu(f):=\sum^n_{i=1}x_i\mu(A_i)\in[0,\infty]}} | * {{MM|1=I_\mu(f):=\sum^n_{i=1}x_i\mu(A_i)\in[0,\infty]}} | ||
Note that this is independent of the particular ''standard representation'' of {{M|f}}. | Note that this is independent of the particular ''standard representation'' of {{M|f}}. | ||
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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].
