403 Definitions

3. measurable space [2, pp. 25]

A measurable space is the space \((X,\mathcal{M})\) if

• \(\mathcal{M} \sub P(X)\)

where

• \(\mathcal{M}\) is \(\sigma\)-algebra
  
• \(X\) is a set
• \(P(X)\) is the family of all subsets [2, pp. 2]

Note: not to be confused with a “measure space.”

3. measure space [2, pp. 25]

A measure space is the space \((X,\mathcal{M}, \mu)\)

where

• \(\mu\) is the measure

Note: not to be confused with a “measurable space.”

2. measurable set [2, pp. 25]

A measurable set is the \(\sigma\)-alegbra, \(\mathcal{M}\), in a measurable space: \[ \mathcal{M} \in (X,\mathcal{M}) \]

1. simple function [1] \[s: X \rightarrow \mathbb{R}\]

\[ s = c_1 \chi_{E1} + c_2 \chi_{E2} \dots + c_n \chi_{En} \]

2. borel measure [2, pp. 33]

…

1. lebesgue measure [2, pp. 37]

• \(\mu_F\) is the complete lebesgue measure
• \(F(x)=x\) function associated with \(\mu_F\)
• \(m\) is the length of the measure
• \(\mathcal{L}\) is the lebesgue measurable set

Use of the lebesgue measure [3, pp. 33]

1. lebesgue integral [2, pp. 56]

..

403.1 References

1. https://e.math.cornell.edu/people/belk/measuretheory/DefiningTheIntegral.pdf
2. [Pure and Applied Mathematics_ A Wiley-Interscience Series of Texts, Monographs and Tracts] Gerald B. Folland - Real Analysis_ Modern Techniques and Their Applications (1999, Wiley-Interscience) - libgen.li.pdf
3. William Feller - An Introduction to Probability Theory and Its Applications, Vol. 2 (1971, Wiley) - libgen.li.pdf

• Definitions