T1: Measure Theory

March 02, 2024

• 1. Why do we need Measure Theory?
• 2. Set
  • Operation on sets
• 3. The Probability Space
• Reference:

1. Why do we need Measure Theory?

When we talk about probability theory, we want to know how to define an event and how to calculate the probability. But before that, we need to know some rules to handle events including how to combine them and not combine them. We need to define collections of sets with certain structures. So, we need Measure Theory.

Also, there is a good example on the page titled Measure theory in probability-Towards rigorous probability.

The first wired thing: when we consider the situation where we throw a tiny ball on the real line of length six, the probability is 0. The second weird thing is that if the probability of a ball hitting any fixed point is zero, then the sum of these probabilities is not one.

There is also an interesting answer in StackExchange [3].

In probability theory, you are concerned with assigning probabilities to events (sets), so you are dealing with functions whose inputs are sets and whose outputs are real numbers. This leads to $\sigma$-algebras and measure theory.

2. Set

Set may be the most important conception in math. There are many famous books on the theory of the set. We say that a set is a collection of ‘objects’ or elements. The basic set operations contain the following.

• Union: \(A \cup B = \{x: x\in A \ or \ x\in B\}\)

• Intersection: \(A \cap B = \{x: x\in A \ and \ x\in B\}\)

• Complement: \(A'= \{x: x\not \in A\}\)

• Difference: \(A\setminus B = A \cap B'\)

• Symmetric Difference: \(A \triangle B = (A\setminus B)\cup (B\setminus A)\).

• The de Morgan formula

  \[{(\bigcup_{k=1}^n A_{k})} '=\bigcap_{k=1}^n{A_{k}}'\]

  and \({(\bigcap_{k=1}^n A_{k})} '=\bigcup_{k=1}^n{A_{k}}'\)

Operation on sets

Let \(\mathcal{A}\) be a non-empty collection of subsets of \(\Omega\).

• \[A\in \mathcal{A} \to A' \in \mathcal{A}\]
• \[A,B\in \mathcal{A} \to A\cup B \in \mathcal{A}\]
• \[A,B\in \mathcal{A} \to A\cap B \in \mathcal{A}\]
• \[A_n \in \mathcal{A}, n\geq 1, \to \bigcup_{n=1}^\infty A_{n}\in \mathcal{A}\]

We say that \(\mathcal{A}\) is an algebra (field) if \(\Omega \in \mathcal{A}\) and (1) and (2) hold, \(\mathcal{A}\) is \(\sigma\)-algebra if \(\Omega\in \mathcal{A}\) and (1), (4) hold.

Math is Beautiful, isn’t it?

3. The Probability Space

Now, we can define the probability space.

The triple \((\Omega, F, P)\) is a probability (measure) space if

• \(\Omega\) is the sample space
• \(F\) is \(\sigma\)-algebra of sets (events)

• \(P\) satisfies Kolmogrow axioms:

  • For any \(A\in F\), there exists a number \(P(A) \geq 0\)
  • \[P(\Omega) = 1\]
  • Let \(\{A_n, n\geq 1\}\) be disjoint, then \(P(\bigcup_{n=1}^\infty A_{n}) = \Sigma_{n=1}^\infty P(A_n)\)

Reference:

1. Probability: A Graduate Course, Allan Gut
2. Measure theory in probability,Towards Data Science, Xichu Zhang
3. Stackexchange: why-measure-theory-for-probability