REFERENCE SHEET

σ-algebras & measures

Not every subset needs to be measurable; specify the collection on which size is defined.

Before this Sets & maps, Completeness & sequences

μ(⋃ₙ Aₙ) = Σₙ μ(Aₙ) for pairwise disjoint measurable Aₙ

Definitions

σ-algebra

A family ℱ of subsets of X containing X and closed under complements and countable unions. It is consequently closed under countable intersections.

Measure

A map μ:ℱ→[0,∞] with μ(∅)=0 and countable additivity on disjoint measurable sets.

Borel and Lebesgue sets

The Borel σ-algebra is generated by open subsets. Lebesgue measure on ℝ extends interval length and is complete: all subsets of null sets are measurable.

Key theorems

Continuity from below

For increasing measurable Aₙ, μ(⋃Aₙ)=lim μ(Aₙ), allowing +∞.

Continuity from above

For decreasing measurable Aₙ with μ(A₁)<∞, μ(⋂Aₙ)=lim μ(Aₙ).

Countable null unions

A countable union of measurable sets of measure zero has measure zero.

Notable applications

Probability

A probability measure is a measure with total mass 1.

Almost-everywhere reasoning

Ignore changes on null sets when studying integration and many function-space properties.

Work it out

Try before revealing

01 / Compute

What is the Lebesgue measure of ℚ∩[0,1]?

Hint

Enumerate the rationals and use singleton measures.

Solution

Each singleton has measure 0 and the rationals are countable. Countable subadditivity gives total measure 0.

02 / Counterexample

Why is finite measure needed for continuity from above?

Hint

Consider Aₙ=[n,∞) in ℝ.