Definition

A σ-algebra \(\Sigma\) on \(A\) is a subset of \({\cal{P} (A)}\) where:

A set in a σ-algebra is called a measurable set.
The "σ-" prefix is used to name things having countable properties. In this case, countable union.

σ-algebra on \(A\)

Measurable set in \(A\)

Measurable space

A measurable space has a set \(A\) and a sigma algebra \(\Sigma\) on \(A\).

Countable intersection in σ-algebras

Let \(\Sigma\) be a σ-algebra. The countable intersection of a sequence of sets is in \(\Sigma\).
Proof:

Maximal σ-algebra

Trivial σ-algebra

Unit σ-algebra

Semi-finite σ-algebra

Trace σ-algebra

Preimage σ-algebra

σ-algebra generated by \(S\)

Generator of \(\Sigma\)

Theorem

Intersection of sigma algebras on X is sigma algebra.

For system G there is a coarsest σ-algebra containing G.

(i) Arbitrary intersection of σ-algerbras is σ-algebra.

(ii) For every system of subsets \(\mathcal{G}\) of \(X\) there is a smallest σ-algebra containing \(\mathcal{G}\).

G is called generator, σ(G) is sigma algebra generated by G.

Self-generated σ-algebra

Unit-generated σ-algebra

Definition

Borel σ-algebras is a bridge between topology and measure theory. A Borel σ-algebra is simultaneously a σ-algebra and a topology on a set \(A\).

Borel σ-algebra

Borel set

Measure

Signed measure

Pre-measure

Measurable space

A measurable space has a set \(A\) and a sigma algebra \(\Sigma\) on \(A\).

Measure space

Finite measure

Probability measure

Probability space

σ-finite measure

A σ-finite measure is a measure where the measure of the full space \({\mu(A)}\) can be \(\infty\), but any other measurable set has finite measure.

σ-finite measure space

Dirac measure

Countability measure

Counting measure

Discrete probability measure

Discrete probability space

Trivial measure

Lebesgue measure

A Lebesgue measure is a measure on \({\cal{B} {\mathbb{R}}}\) defined by \({\mu(\x..i{a}{b)} = b - a}\)

Tail set of a sequence

Borel–Cantelli

F

Proposition 4.3

...

Remark

(Premeasures)

Theorem

Lebesgue measure exists and is uniquely determined by its values on the Borel sets.

Invariant under translations, rotations and reflections.

... invertible matrix ...

Lemma

Additive + Continuous from below = Measure

Lemma

Additive + Continuous = Measure

Definition

...

Measurable mapping

Measurable function

Composition

Let \(f\) and \(g\) be measurable. Then \({f \circ{} g}\) is measurable.

Level-constant function

A level-constant function (or simple function) is a function that is constant on measurable intervals. The image takes on a set of discrete values, one for each interval.

Lebesgue sum of a simple function

Lebesgue integral of a simple function

Definition

k

Lebesgue integral with respect to \(\mu\)

Lebesgue integral

..

\({L}^{1}\) integrable function

Lebesgue integrable function

Lebesgue vs Riemann integration

In Riemann integration, the domain is being partitioned. In Lebesgue integration, the codomain is partitioned.

Integration almost everywhere

Let \(f\) and \(g\) be measurable functions where \(f(x) = g(x)\) almost everywhere. Then their integrals are identical.

Integral wrt. pushforward measure

Integral of a measurable function

Monotone convergence theorem

..

Fatou’s lemma

Reverse Fatou’s lemma

Fatou–Lebesgue theorem

Dominated convergence theorem

Let \(E\) be a measure space and \({f}_{{1:\infty}}\) a sequence of measurable functions.

Measure theory

σ-algebras

Borel σ-algebras

Measures

Borel–Cantelli

Measurable Mappings

Lebesgue integration

Convergence theorems

Types of integrals

Radon–Nikodym