desirable properties for measure

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Define measure of subsets of $R$ , which we will eventually call Lebesgue measure.

Properties we want for this idea

$m (E)$ is defined for all $E \subset R$
if $I$ is an interval then $m (E) = ℓ (I)$ the "length" of $I$
If ${E_{n}}$ is a (countable) collection of disjoint subsets of $E$ such that $E = ⋃_{n} E_{n}$ , then we want $m (⋃_{n} E_{n}) = m (E)$
Translation invariance. ie, if $E \subset R$ and $x \in R$ , then $m (x + E) = m ({x + y | y \in E}) = m (E)$

SPOILER

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

Instead, we can drop the assumption that measure is defined on all subsets of $R$ . 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 $m^{*} : P (R) \to [0, \infty)$ (which is defined on all of $R$ ) satisfying (2), almost (3), and (4).
Then, restrict $m^{*}$ to well-behaved subsets of $R$
^spoiler

References

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desirable properties for measure

References

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