Applied Probability - F26 - Lecture 3

Overview

Arash Amini develops the pi-lambda theorem as a practical uniqueness tool: two probability measures that agree on a generating pi-system agree on its generated sigma-algebra. He then explains measure construction through algebras, premeasures, and the Carathéodory outer-measure method, before characterizing probability measures on R by right-continuous CDFs and introducing random variables as measurable maps.

Key takeaways

If two probability measures agree on a pi-system that generates a sigma-algebra, the pi-lambda theorem guarantees that they agree on the entire sigma-algebra.
A CDF F(x) = P((−∞, x]) uniquely determines a probability measure on R because the half-lines form a generating pi-system.
Measure construction begins with a premeasure on an algebra; countable additivity is required for disjoint unions only when their union remains in that algebra.
Carathéodory’s method builds an outer measure from infimum-cost covers, then identifies measurable sets by the additive splitting condition.
A probability distribution function must be nondecreasing, right-continuous, and approach 0 and 1 at the two ends of the real line.
Random-variable events are preimages: measurability ensures that {X ∈ B} is an event in the source sigma-algebra and can therefore be assigned a probability.

Chapters

0:00 Why Agreement on a Generating Collection Is Not Enough

Two measures can agree on a collection that generates a sigma-algebra yet differ on other sets in that sigma-algebra.
The sets where two measures agree form a lambda-system: it contains the whole space, is closed under complements, and is closed under countable disjoint unions.
Lambda-systems have weaker closure requirements than sigma-algebras, so an additional condition is needed to prove uniqueness.

3:00 Pi-Systems and the Pi-Lambda Theorem

A pi-system is closed under pairwise intersections, and hence under finite intersections.
Half-lines and suitable collections of half-open intervals are examples used to generate the Borel sigma-algebra on R.
The pi-lambda theorem states that if a pi-system P is contained in a lambda-system L, then the sigma-algebra generated by P is also contained in L.

10:00 The Core Idea Behind the Pi-Lambda Proof

Consider the smallest lambda-system containing P; the proof shows that this smallest system is also closed under intersections.
The intersection argument extends the known closure for sets in P to pairs of sets in the generated lambda-system.
A collection that is both a lambda-system and a pi-system is a sigma-algebra, which yields the theorem.

14:00 Proving Uniqueness of Probability Measures

For probability measures P and Q, the sets A satisfying P(A) = Q(A) form a lambda-system.
Agreement on complements follows from P(Aᶜ) = 1 − P(A) and Q(Aᶜ) = 1 − Q(A); countable disjoint additivity supplies the other key closure property.
If P and Q agree on a generating pi-system, the pi-lambda theorem gives agreement throughout its generated sigma-algebra.

21:00 CDFs Determine Probability Measures on the Real Line

The half-lines (−∞, x] form a pi-system that generates the Borel sigma-algebra on R.
A probability measure is therefore determined by its distribution function F(x) = P((−∞, x]).
F is a real-valued function that encodes a set function on the much larger collection of Borel subsets of R.

25:00 Lebesgue Measure Uniqueness and the Need for Existence

Agreement on intervals can determine a measure when the intervals form an appropriate generating pi-system.
Lebesgue measure on R is characterized by assigning length b − a to intervals with endpoints a and b, alongside the required finite-measure or sigma-finite conditions.
Uniqueness does not establish that a measure with prescribed interval values exists; additivity and continuity can impose consistency constraints.

32:00 Algebras as Simpler Domains for Defining Measures

An algebra of sets contains the whole space and is closed under complements and finite unions; a sigma-algebra additionally requires countable-union closure.
The interval algebra on R consists of finite unions of intervals and is simpler to describe than the Borel sigma-algebra it generates.
Specifying values on an algebra provides a manageable starting point for constructing a measure on the generated sigma-algebra.

36:00 Premeasures and Countable Additivity on an Algebra

A premeasure μ₀ assigns zero to the empty set and is countably additive for disjoint sequences whose union remains in the algebra.
The definition does not require every countable union to belong to the algebra; it requires additivity whenever that union does belong.
For interval lengths, countable decompositions must give values consistent with the length of the original interval.

46:00 Extending a Premeasure with Outer Measure

A premeasure on an algebra extends to a measure on the sigma-algebra generated by that algebra, under the extension theorem discussed in class.
For a general set, define an outer measure by covering it with algebra sets and taking the infimum of the sums of their premeasure values.
Carathéodory measurability requires every set S to split additively across E: μ*(S) = μ*(S ∩ E) + μ*(S \ E).

53:00 Carathéodory-Measurable Sets and Lebesgue Measure

The sets satisfying the Carathéodory splitting condition form a sigma-algebra, and restricting the outer measure to them gives a measure.
For Lebesgue measure, start with interval length on an interval algebra; verifying the premeasure property allows extension to the Borel sigma-algebra.
The resulting Lebesgue-measurable sets include Borel sets and additional subsets of null sets, extending beyond the Borel domain.

59:00 Necessary Properties of a Probability Distribution Function

If F(x) = μ((−∞, x]) for a probability measure on R, then F is nondecreasing and right-continuous.
The probability limits are F(x) → 0 as x → −∞ and F(x) → 1 as x → +∞.
Interval probabilities follow from F: μ((a, b]) = F(b) − F(a), and the mass at x is F(x) − F(x−).

1:03:00 Constructing Measures from Right-Continuous CDFs

A nondecreasing, right-continuous function with limits 0 and 1 defines a probability measure on the Borel sets of R.
One construction assigns F(b) − F(a) to intervals (a, b] and extends the resulting premeasure from the interval algebra.
Right continuity is essential to verifying the premeasure property; an alternative construction can start from Lebesgue measure.

1:05:00 A Mixed Distribution: an Atom and a Uniform Component

The example combines one-half of a point mass δ₀ with one-half of Lebesgue measure restricted to the interval from 1 to 2.
Its CDF jumps by one-half at zero, stays flat between zero and one, rises linearly from one to two, and then remains at one.
The distribution has an atom and a continuous component, so it is neither purely discrete nor described by a density alone.

1:10:12 Random Variables as Measurable Maps

A random variable is a measurable function from a probability space (Ω, F) to a target measurable space; a real-valued variable uses the Borel sigma-algebra on R.
For the six-face die, the indicator of an odd outcome maps outcomes 1 through 6 to values 1 or 0.
An event such as {X ∈ B} is the preimage X⁻¹(B); measurability requires this preimage to belong to F for every measurable target set B.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, ARASH AMINI.