lacunary - Mathnotes

Outer Measure

Our intention is to come up with an @extension of the length of intervals that works on more complicated domains than the @subintervals that @Riemann integration uses, for example, unions of subsets of the reals.

Definition: length of open interval Notation: ℓ \@{length-of-open-interval}

The length ℓ(I) of an @open interval I is defined by

{b−a if I=(a,b) for some a,b∈R with a<b,0 if I=∅,∞ if I=(−∞,a) or I=(a,∞) for some a∈R,∞ if I=(−∞,∞)

Referenced by (2 direct, 1 transitive)

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Definition: outer measure \@{outer-measure}

The outer measure |A| of a set A⊂R is defined by

|A|=inf{∑k=1∞ℓ(Ik):I1,I2,… are open intervals such that A⊂⋃k=1∞Ik}.

Referenced by (2 direct)

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Properties of Outer Measure

TODO: I have proofs for these written down in paper notes, transcribe them.

Theorem: Countable Sets Have Outer Measure 0 \@{countable-sets-have-outer-measure-0}

Every countable subset of R has outer measure 0.

Corollary \@{corollary-4}

From this and The set of rational numbers is countable., we have that |Q|=0.

Theorem: Outer Measure Preserves Order \@{outer-measure-preserves-order}

Suppose A and B are subsets of R with A⊂B. Then |A|≤|B|.

Definition: translation \@{translation}

If t∈R and A⊂R, then the translation t+A is defined by

t+A={t+a:a∈A}.

Theorem: Outer measure is translation invariant \@{outer-measure-is-translation-invariant}

Suppose t∈R and A⊂R. Then |t+A|=|A|.

Theorem: Countable subadditivity of outer measure \@{countable-subadditivity-of-outer-measure}

Suppose A1,A2,… is a sequence of subsets of R. Then

|⋃k=1∞Ak|≤∑k=1∞|Ak|.

Theorem: Outer measure of a closed interval \@{outer-measure-of-a-closed-interval}

Suppose a,b∈R with a<b, Then |[a,b]|=b−a.

Note \@{outer-measure-of-a-closed-interval-note}

The proof for this theorem uses Heine-Borel.

Theorem: Nontrivial intervals are uncountable \@{nontrivial-intervals-are-uncountable}

Every interval in R that contains at least two @distinct elements is uncountable.

Now, the problem with outer measure, or really, the problem with R.

Theorem: Nonadditivity of outer measure \@{nonadditivity-of-outer-measure}

There exist @disjoint subsets A and B or R such that

|A∪B|≠|A|+|B|.

Note \@{nonadditivity-of-outer-measure-note}

The proof of this theorem uses Vitali Sets.

Referenced by (1 direct)

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