Here are my notes when reading about σ-algebras (sigma-algebra).

### Motivation

**Example**: Toss a coin, have it come up heads, and if it comes up

tails, pick a random number in the range [0,1]. How do you come up

with a PDF/CDF?

**Further reading:**Look at the excerpt from A User's Guide to Measure Theoretic Probability.

**Related video**: Why use measure theory for probability?

So, measure-theoretic probability unified the discrete and continuous cases.

### First approach

Pick a set of

flipping coins, this set is {H,T}, and when rolling dice this set

is {1,2,3,4,5,6}.

Give a function \(P\) that

subsets of Ω. You can write this as: \(P : 2^{\Omega} \to [0,1]\).

The \(2^X\) notation denotes the power set of \(X\).

We'd like this function to behave

\(P(\text{roll even number}) = P(\{2,4,6\})\). If \(P\) assigned

a number to

However, for the continuous case, something like \(2^{[0,1]}\) leads us to paradoxes, so the question is: \(P : ? \to [0,1]\).

**all possible outcomes**Ω (capital omega). Whenflipping coins, this set is {H,T}, and when rolling dice this set

is {1,2,3,4,5,6}.

Give a function \(P\) that

**assigns a probability**to arbitrarysubsets of Ω. You can write this as: \(P : 2^{\Omega} \to [0,1]\).

The \(2^X\) notation denotes the power set of \(X\).

We'd like this function to behave

*nicely*and obey some intuitive laws:- \(P(\Omega)=1\)
- For every event \(A \in \Omega\), \(P(A) \ge 0\)
- For any two disjoint events \(A,B\), \(P(A \cup B)=P(A)+P(B)\)

\(P(\text{roll even number}) = P(\{2,4,6\})\). If \(P\) assigned

a number to

*all*subsets, then we'd be happy.However, for the continuous case, something like \(2^{[0,1]}\) leads us to paradoxes, so the question is: \(P : ? \to [0,1]\).

### Sigma-algebras

Choose some subset of the power set and define the domain of \(P\) using it:

$$ P : \Sigma \to [0,1] $$ $$ \emptyset \le \Sigma \le \Omega $$

What we'd like Σ to contain:

$$ \Sigma = \{ \emptyset, \Omega \} $$

There are certain rules a σ-algebra has to obey:

Note: To resolve the paradox mentioned above, see the Borel σ-algebra.

In the usual parsimonious fashion, the above three rules along with

the definition of \(P\) in the previous section, are used to derive

the rest.

$$ P : \Sigma \to [0,1] $$ $$ \emptyset \le \Sigma \le \Omega $$

What we'd like Σ to contain:

- The empty set ∅, so that we can talk about the

probability of something**not**happening. - The set of all possible outcomes Ω, so we can talk about the

possibility of anything happening.

**trivial σ-algebra**:$$ \Sigma = \{ \emptyset, \Omega \} $$

There are certain rules a σ-algebra has to obey:

- \(\Omega \in \Sigma\): As above
- \(A \in \Sigma\) implies that the complement \(A^C \in \Sigma\): if I can talk

about some event A happening, then I'd like to talk about some

event A**not**happening - \(A,B \in \Sigma\) implies that \(A \cup B \in \Sigma\)

*countable unions*with reference to (3) above.Note: To resolve the paradox mentioned above, see the Borel σ-algebra.

In the usual parsimonious fashion, the above three rules along with

the definition of \(P\) in the previous section, are used to derive

the rest.

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