202405211543
Status: #idea
Tags: Probability Theory

Continuous Probability Spaces

Continuous Probability Spaces are to Borel-Sigma Algebras what Discrete Probability Spaces are to $2^{\Omega }$.

Make sure you have a good understanding of Borel$-\sigma -$algebras before proceeding.

In continuous probability spaces, our Measurable Space is by default $(\Omega ,\mathcal{B}(\Omega ))$ which we typically just write $(\Omega ,\mathcal{B})$.

In general we choose our $\Omega$ to be $(0,1]$ where we expressly exclude $0$ and include $1$, while this might look arbitrary we do so for two reason:

• Any results we can derive for this interval can be generalized for sample spaces $(0,1),[0,1],[0,1)$
• It makes some upper level derivations easier and we avoid issues like $log(0)$ and whatnot.

Defining the Uniform Distribution

We want two things to do that:

• We want our measure of the probability to be proportional to the length of the interval under consideration
• We want our measure of the probability to be invariant under translation, any arbitrary of length $\alpha$ (where $\alpha =|b-a|$) should have the same Probability Measure (According to Kolmogorov) no matter where they are.

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