Last lesson we talked about measurable sets.
This lesson we will talk about constructing a \sigma-field of sets, called the Borel \sigma-field, and then we will talk about how to construct a measure on it, called the Lebesgue measure.
119.1 Lesson 2: Borel Sets
This video, “Measure Theoretic Probability, Lesson 2,” focuses on \sigma-fields and introduces the concept of Borel sets.
The presenter begins by reviewing the definition of a sigma field as a collection of sets that must contain the full set \Omega, be closed under complements, and be closed under countable unions.
She then introduces the idea of a \sigma-field generated by a collection of sets \mathcal{A}, defining it as the smallest \sigma-field that contains all the sets in that collection.
Key points covered include:
- Intersection of Sigma Fields The video proves that the intersection of two sigma fields is also a sigma field. This property is crucial for understanding the formal definition of a generated sigma field.
- Containment of Generated Sigma Fields: It’s demonstrated that if a collection of sets ‘a’ is contained within another collection ‘b’, then the sigma field generated by ‘a’ is contained within the sigma field generated by ‘b’.
- Examples of Generated Sigma Fields: The video provides trivial examples of sigma fields generated by a single set or by an already existing sigma field.
- Borel Sigma Field and Borel Sets: The most important example introduced is the Borel sigma field, generated by all open intervals on the real line. The video then shows how other types of intervals (half-open, closed, and singleton sets) can be constructed within the Borel sets , and that the Borel sets can be generated by other starting collections of intervals.
- The video concludes by emphasizing the importance of Borel sets in real analysis and measure theoretic probability, hinting that the next video will define the concept of a measure.
119.1.1 Recap!
We begin with a bit of a recap of the last lesson.
- Measure = a function that takes in a set, and give a number as output
- Lebesgue measure = function that gives outer-length/area/volume of a set
At the end of the last lesson we saw a set that was not a measurable set. This happened because it was not closed under countable unions. Jen raised the following question:
if we start with a collection of sets, can we generate a \sigma-field from it by simply adding in the sets that we need to make it closed under complements and countable unions?
The next part of a probability space is \mathcal{A} which is a collection of subsets of \Omega that we can measure.
Definition 119.1 (\sigma-field generated by a collection of sets)
Given a collection of sets \mathcal{A}, the \sigma-field generated by \mathcal{A} is the smallest \sigma-field that contains \mathcal{A}.
119.2 Notation: \sigma(\mathcal{A}).
if \mathcal{F} is a \sigma-field that contains \mathcal{A}, then it follows that \sigma(\mathcal{A}) \subseteq \mathcal{F}.
closed under finite unions
