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May 1st, 2018, 05:06 AM  #1 
Senior Member Joined: Nov 2011 Posts: 247 Thanks: 3  Integration Axioms
What are the axioms of Integration?

May 1st, 2018, 05:34 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 534 Thanks: 306 Math Focus: Dynamical systems, analytic function theory, numerics 
1. The real numbers are a closed ordered field. 2. Thou shalt not covet thy neighbor's wife. 3. He who smelt it dealt it. Note: If in addition you accept the axiom of choice, then 3 is equivalent to: 4. He who said the rhyme, did the crime. 
May 1st, 2018, 12:05 PM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 
Typically, the term "axiom" refers to a "mathematical structure", such as the "integers", "real numbers", and "functions of real numbers". Integration however is a operation applied to the mathematical structure of functions on the real numbers. "Integration", itself, is not a mathematical structure so the word "axiom" does not apply.

May 1st, 2018, 12:08 PM  #4  
Senior Member Joined: Aug 2012 Posts: 2,134 Thanks: 621  Quote:
 
May 1st, 2018, 12:27 PM  #5 
Senior Member Joined: Oct 2009 Posts: 688 Thanks: 223 
It is in fact possible to give axioms for integration. So called Daniell integration theory does this and is an alternative way to construct the Lebesgue integral and Lebesgue measure. Given are a set $X$, a set $\mathcal{U}$ consisting of function $X\rightarrow \mathbb{R}$, and a function $I:\mathcal{U}\rightarrow \mathbb{R}$. The following axioms are considered:  $\mathcal{U}$ is a vector space for the usual operations on functions $X\rightarrow \mathbb{R}$  For any $f,g\in \mathcal{U}$, we have that $\sup(f,g)$ and $\inf(f,g)$ are also in $\mathcal{U}$.  $I$ is a linear function  If $f\geq 0$, then $I(f)\geq 0$  If $f_n\in \mathcal{U}$ for every $n\in \mathbb{N}$ is such that $$f_1\geq f_2 \geq f_3 \geq ...\geq 0$$ and if $f_n\rightarrow 0$ pointswise, then $I(f_n)\rightarrow 0$. 
May 1st, 2018, 12:27 PM  #6 
Senior Member Joined: Oct 2009 Posts: 688 Thanks: 223 
It is in fact possible to give axioms for integration. So called Daniell integration theory does this and is an alternative way to construct the Lebesgue integral and Lebesgue measure. Given are a set $X$, a set $\mathcal{U}$ consisting of function $X\rightarrow \mathbb{R}$, and a function $I:\mathcal{U}\rightarrow \mathbb{R}$. The following axioms are considered:  $\mathcal{U}$ is a vector space for the usual operations on functions $X\rightarrow \mathbb{R}$  For any $f,g\in \mathcal{U}$, we have that $\sup(f,g)$ and $\inf(f,g)$ are also in $\mathcal{U}$.  $I$ is a linear function  If $f\geq 0$, then $I(f)\geq 0$  If $f_n\in \mathcal{U}$ for every $n\in \mathbb{N}$ is such that $$f_1\geq f_2 \geq f_3 \geq ...\geq 0$$ and if $f_n\rightarrow 0$ pointswise, then $I(f_n)\rightarrow 0$. 

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