Undergraduates with some exposure to measure theory may know that the Lebesgue measure is the unique, up to multiplication by a constant, translation-invariant Borel measure. If not, then no worries, as today’s post will cover the proof of this result. My motivation for selecting this topic for a blog post was I was trying to remember the proof that Lebesgue measure is invariant under transformation by an orthogonal matrix. The proof of this invariance result, along with a more general result about the behavior of the Lebesgue measure under linear transformations, will be proven in a later post.

Let be a set. A family is a *Dynkin system* if

- pairwise disjoint .

It is immediate from the definition that , and is closed under finite disjoint union. It is evident that every -algebra is a Dynkin system, but the following example shows the converse is false:

Let for some fixed . Set . That is a Dynkin system contains and is closed under countable disjoint union is obvious. For closure under relative complement, is the difference of two even numbers and therefore even. Evidently, is not a -algebra, since , but .

Proposition 1.Let . Then there is a smallest (also minimal, coarset) Dynkin system containing . is called the Dynkin system generated by . Moreover, .

*Proof. *Define to be the intersection of all Dynkin systems containing . Then exists and is non-empty since is a Dynkin system and . That is minimal is immediate from the definition. Since every -algebra is a Dynkin system and , we have that .

Lemma 2.A Dynkin system is a -algebra if and only if it is closed under finite intersection: .

*Proof. *If is a -algebra, then *a fortiori* it is closed under finite intersection. Suppose is closed under finite intersection. Let be an arbitrary sequence in . Define

By our -stable hypothesis, for each , and is pairwise disjoint. Hence, . So is closed under countable union, and therefore a -algebra.

Proposition 3.If is stable under finite intersections, then .

*Proof. *Tautologically, , so by definition of , it suffices to show that is a $\sigma$-algebra. By the preceding lemma, it suffices to show that is stable under finite intersection. Fix and define the family

I claim that is a Dynkin system. . Observe that if , then

by the Dynkin axioms applied to . Closure under countable disjoint union is obvious. Observe that since is -stable, . Since is a Dynkin system, we have that , when . Hence,

which is equivalent to saying that is closed under finite intersections.

The proof of the following proposition is obvious from our preceding work.

Proposition 4.Let be a -algebra, be a Dynkin system, and be two collections of subsets of . Then

The next proposition is a useful tool for checking the equivalence of two measures on a given measurable space.

Proposition 5.(Uniqueness of Measures) Assume that is a measurable space and that is generated by a family such that

- There exists an exhausting sequence with
Then any two measures that coincide on and satisfy are equal on (i.e. ).

*Proof. *Let be two measures that coincide on and satisfy for each . For each , define

I claim that is a Dynkin system. Clearly, , and if , then by additivity,

If is pairwise disjoint, then using that ,

Obviously by hypothesis that is -stable and that agree on . Hence, for each . Since is -stable, by Proposition 3, . Hence, . Since , for every ,

We denote the Lebesgue measure on by .

Theorem 6.(Lebesgue Translation-Invariance)The -dimensional Lebesgue measure is invariant under translations:

Every measure on which is invariant under translations and satisfies is a real multiple of the Lebesgue measure: .

*Proof. *We first verify that is indeed a Borel set. Define

I claim that is a -algebra. It is clear that , and is closed under countable union. For closure under relative complement, I claim that . Indeed,

Since translation is a homeomorphism, we have that , where is the collection of open (in the Euclidean topology) subsets of . By definition, , which implies that .

For , set for fixed. It is clear that is a measure on . Let denote the collection of half-open rectangles in . Let and observe that

Hence,

which shows that . Since is -stable, generates , and has an exhausting sequence , we have by Proposition 5 that .

Let be a measure on which satisfies and is translation invariant. Observe that, if is a positive integer, then by additivity of , we see that

,

which implies that . If are positive integers, then

where we use the additivity of measures. It follows from the translation-invariance of that, for half-open rectangle as above, but with for , the formula

holds. By Proposition 5, .

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