Measures

Classes of sets

`Omega != O/`, `P(Omega) = {A: A sub Omega}`, `ccF sub P(Omega)`

`pi` system
`A, B in ccF => A nn B in ccF`	If `C` is a `pi`-system, then `lambda( C ) = sigma(: C :)`. If `C` is a `pi`-system, and `ccL` is a `lambda`-system containing `C` then `ccL sup sigma(: C :)`
Semi-algebra
`A, B in C => A nn B in C` `AA A in C, A^c = uu_(i=1)^(k	The smallest algebra containing a semi-algebra is the class of finite unions of sets from C.
Algebra
`Omega in ccF` `A in ccF => A^c in ccF` `A, B in ccF => A uu B in ccF`	An intersection of `sigma`-algebras is always a `sigma`-algebra, but the union may not even be an algebra. If `sfF sub P(Omega)` then the `sigma`-algebra generated by `ccF`, `sigma(: ccF :) = ` intersection of all `sigma`-algebras containg `ccF`.
`sigma`-Algebra
`ccF` is an algebra closed under countable (monotone) unions	A useful class of `sigma`-algebra are those generated by open sets of a topological space. A topological space is a pair `(S,T)` where `S` is a non-empty set and `T` is a collection of subset st. `S in T`, and `T` is closed under intersection and uncountable unions. Elements of `T` are called open sets. The Borel `sigma`-algebra on a topological space `S` (in particular a metric or Euclidean space) is defined as the `sigma`-algebra generated by the collection of all open sets in `S` = `sigma(:T:)`.
`lambda` system
`Omega in ccF` `A, B in ccF => B \\ A in ccF` `A_n in ccF, A_n sub A_(n+1) AA n>= 1 =>` `U_(n>=1) A_n in ccF`	Every `sigma`-algebra is a `lambda`-system. The intersection of two `lambda`-systems is a `lambda`-system `lambda( C )` defined similarly to `sigma(:C:)`

Measures

A set function is an extended real-valued function on a class of subsets of `Omega`. A set function `mu : A -> [0, oo]` is a measure if:

• `mu(O/) = 0`
• countably additive or monotone continuous from below.

A measure is infinite if `mu(Omega) = oo`, finite otherwise. A measure with `mu(Omega) = 1` is a probability measure. A measure is `sigma`-finite if there is a countable collection of sets such that `uu A_n = Omega` and `mu(A_n) < oo`.

Common measures:

• counting measure: `mu(A) = |A|`, `mu` is finite iff `Omega` is finite, `sigma`-finite if `Omega` is countable
• discrete probability mass: point mass on integers
• Lebesgue-Stieltjes measures on `RR`: arise from any nondecreasing function `F: RR -> RR`, such that `mu_F (a,b] = F(b+) - F(a+)`, all `sigma`-finite.

A measure `mu` on an algebra `F` has the following properties:

• mononticity: `A sub B => mu(A) <= mu(B)`
• finite subadditivity: `mu(A_1 uu .. uu A_k) <= mu(A_1) + ... + mu(A_k)`
• inclusion exclusion formula: `mu(A uu B) = mu(A) + mu(B) - mu(A nn B)`
• monotone continuity from above if `mu(A_(n_0)) < oo` for some `n_0 in NN`
• countable subadditivity

Uniqueness of measures: `mu_1, mu_2` two measures on a measurable space `(Omega, F)`. Let `C sub F` be a `pi`-system st. `F = signa(:C:)`. If `mu_1(B) = mu_2(B) AA B in C` then `mu_1(A) = mu_2(A) AA A in F`.

The extension theorems and Lebesgue-Stieltjes measures

A set function `mu: C -> [0, oo]` defined on a semi-algebra is a measure if:

• `mu(O/) = 0`
• countably additive where possible

Given a measure on a semi-algebra `C`, the outer measure induced by `mu` is the set function `mu^**`, defined on `P(Omega)` defined as `mu^**(A) = inf{ sum mu(A_n) : {A_n}_{n>=1} sub C, A sub uu_{n>=1} A_n}`. (approximate `mu` as the least upper bound of the measure of all possible covers). This is exact on `C` and `A`. A set `A` is exactly `mu^**` measurable if `mu^**(E) = mu^**(E nn A) + mu^**(E nn A^c) AA E sub Omega`.

`mu^**` satisfies: (any set functions with these properties is called an outer measure)

• `mu^**(O/) = 0`
• monotonicity: `A sub B => mu^**(A) <= mu^**(B)`
• countable subadditivity

Let `mu^**` be an outer measure on `Omega`. Let `ccM = ccM_{mu^**} = {A: A " is " mu^** "measurable"}`. Then:

• `ccM` is a `sigma`-algebra
• `mu^**` restricted to `ccM` is a measure
• `mu^**(A) = 0 => P(A) sub ccM`

Caratheodory's extension theorem*: Let `mu` be a measure on a semialgebra `C` and `mu^**` be defined as above. Then:

• `mu^**` is an outer measure
• `C sub M_{mu^**}`
• `mu^** = mu` on `C`

`A in M_(mu^**)`, `mu^**(A) < oo`, then `AA epsi >0 EE B_1, B_2, ..., B_k in C, k < oo`, `B` mutually disjoint, s.t. `mu^**(A o+ U B_j) < epsi`, where `E_1 o+ E_2` is the symmetric difference. That is measure, any measure on R can be apprommixtaely by a finite number of intervals, or every `mu^**` measurable set of finite measure is nearly a finite union of disjoint elements from the semialgebra C.

Lebesgue-Stieltjes measures on R

Let `F: RR -> RR` be nondecreasing. Let `C = { (a,b]: -oo <= a <= b < oo} uu { (a, oo): -oo <= a < oo}` and define `mu_F (a,b] = F(b+) - F(a+)`, then:

• `C` is a semialgebra
• `mu_F` is a measure on `C` (requires Heine-Borel theorem)
• `(RR, M_{mu_F^**}, mu_F^**)` is the complete measure space constructed with the Caratheodory extension method, and is called the Lebesgue-Stieltjes measure generated by `F`.

`F(x) = x` is known as the Lebesgue measure.

A Radon measure is finite on bounded intervals. `mu` is a radon measure iff it is a Lebesgue-Stieltjes measure.

Completeness of measures

A measure space is called complete if `AA A in F, m(A) = 0 => P(A) sub F`. The measure space generated by the exetension theory is complete.

It is always possible to complete an incomplete measure: Let `(Omega, F, mu)` be a measure space. Let `barF = {A : B_1 sup A sup B_2}` for some `B_1, B_2 in F` st `mu(B_1 \ B_2) = 0`. For any `A in barF` set `bar mu(A) = bar mu(B_1) = bar mu(B_2)`. Then:

• `barF` is a `sigma`-algebra and `F sub barF`
• `bar mu` is well defined
• `(Omega, barF, bar mu)` is a complete measure space and `bar mu = mu "on" F`.