A **probability measure** gives probabilities to a sets of experimental outcomes (events). It is a function on a collection of events that assigns a probability of 0 and 1 to every event, meeting certain conditions.

The role of a probability measure is to quantify the likelihood of each outcome from each event in the σ-algebra (a collection of sets that have certain properties); Simply put, σ-algebra assigns a probability to an event (e.g. “It will rain today”) and to that event’s complement (e.g. “It will not rain today”) and the union (“It will rain today, or it won’t”) [2].

## Probability Measure Examples

For a **roll of one six-faced die**, the sample space (Ω) = {1, 2, 3, 4, 5, 6}. If A = {1, 3, 5} is the event that the roll is odd, then **P**(A) = ½.

**Uniform Continuous Example**: Let’s say you wanted to pick a random number between 0 and 1, where all numbers are equally likely. The sample space is the closed interval [0, 1]; The probability for any interval within [0, 1] is the length of I. Generally speaking, the uniform probability measure on [a, b] can be defined as [3]:

## Formal Definition of Probability Measure

A **probability measure** **P** on *F* is a real-valued function **P** on *F* with three properties [2]:

**P**(A) ≥ 0, for A ∈*F***P**(Ω) = 1,**P**(∅) = 0,- For the disjoint sequence of events (i.e. A
_{i}∩ A_{j}= ∅ for i ≠ j) then:

- ∅ = the empty set,
- ∩ = intersection,
- ∪ = union,
- A
^{c}= complement (the set of outcomes in Ω which are not in A), - ∈ = is an element of (is in the set),
*F*= a collection of subsets of Ω- σ-field = A collection of subsets of Ω is a σ-field if:
- Ω ∈
*F* - A ∈ F → A
^{c}∈*F* - A
_{n}∈*F*for n = 1, 2, 3, … ∪ (n = 1→∞) A_{n}∈*F*

- Ω ∈

Where:

## References

[1] Probability. Retrieved March 11, 2021 from: https://cims.nyu.edu/~cfgranda/pages/DSGA1002_fall15/material/probability_1.pdf

[2] Kennedy, T. Probability measure and random variables. Retrieved March 11, 2021 from: https://www.math.arizona.edu/~tgk/mc/prob_background.pdf

[3] Probabilities. Retrieved March 11, 2021 from: https://www.math.arizona.edu/~tgk/464_10/chap1_8_26.pdf