desirable properties for measure

[[concept]]

Topics

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Define measure of subsets of $\mathbb{R}$, which we will eventually call Lebesgue measure.

Properties we want for this idea

1. $m(E)$ is defined for all $E\subset \mathbb{R}$
2. if $I$ is an interval then $m(E)=\ell (I)$ the “length” of $I$
3. If $\{E_n\}$ is a (countable) collection of disjoint subsets of $E$ such that $E={\bigcup }_n{E}_n$, then we want $m\left({\bigcup }_n{E}_n\right)=m(E)$
4. Translation invariance. ie, if $E\subset \mathbb{R}$ and $x\in \mathbb{R}$, then $m(x+E)=m(\{x+y|y\in E\})=m(E)$

SPOILER

Unfortunately, this is impossible. Such an $m:\mathcal{P}(\mathbb{R})\to [0,\infty )$ does not exist.

• Instead, we can drop the assumption that measure is defined on all subsets of $\mathbb{R}$. Instead, we can satisfy properties 2-4 and get it defined on “a large collection” of sets, which we will call Lebesgue measurable sets.
• We will construct a function we call the outer measure $m^{\ast }:\mathcal{P}(\mathbb{R})\to [0,\infty )$ (which is defined on all of $\mathbb{R}$) satisfying (2), almost (3), and (4).
• Then, restrict $m^{\ast }$ to well-behaved subsets of $\mathbb{R}$

References

References

See Also

Mentions

Mentions

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