Prove that if m^*(A ΔB)=0, then m^*(A)=m^*(B).

Name: Problem 5, Chapter 2: Measure, Integral and Probability
Uploaded: 2021-08-13T18:46:10-08:00
Duration: 1 min 16 s
Channel: Carson Merrill
Description: Prove that if m^*(A ΔB)=0, then m^*(A)=m^*(B).

Key Concepts

Outer Measure

An outer measure is a function defined on all subsets of a given space that assigns a non-negative number reflecting the 'size' or 'length' of a set, and it satisfies countable subadditivity along with monotonicity. Outer measures are used to extend the notion of measure to more general sets, serving as a precursor to constructing complete measures like the Lebesgue measure.

Symmetric Difference

The symmetric difference between two sets is the set of elements that belong to one of the sets but not to both. Mathematically, it is expressed as A ? B = (A \ B) ? (B \ A). This concept is crucial in measure theory because if the symmetric difference of two sets has measure zero, then the sets are considered to be equivalent in terms of measure.

Measure Zero Sets

A set of measure zero, also known as a null set, is a set that can be covered by countably many intervals of arbitrarily small total length. In measure theory, if the outer measure of a set is zero, its contribution to the measure is negligible, and this property is important when considering almost everywhere equivalences between functions or sets.

Monotonicity of Outer Measure

Monotonicity is a key property of outer measures stating that if one set is a subset of another, then the measure of the smaller set is less than or equal to that of the larger set. This property ensures that the measure behaves consistently with the inclusion relations of sets and is essential in proving equality of measures under certain conditions.

Subadditivity

Subadditivity is the property that the outer measure of a union of any countable collection of sets is less than or equal to the sum of the outer measures of the individual sets. This fundamental property is used to control the measure of complex sets by breaking them into simpler parts, which is a critical tool in many measure theoretic arguments.

Please give Ace some feedback