English 01:

A probability space is a fundamental concept in probability theory, providing a formal mathematical framework for dealing with experiments and random phenomena. It consists of three main components: a sample space, a set of events, and a probability measure. Together, these components allow for a rigorous definition of probability and the ability to compute the likelihood of various outcomes or events.

1. **Sample Space ($\Omega$)**: This is the set of all possible outcomes of a random experiment. Each outcome is called a sample point. For instance, if you roll a six-sided die, the sample space would be $\Omega = {1, 2, 3, 4, 5, 6}$.

2. **Set of Events ($\mathcal{F}$)**: An event is a subset of the sample space and represents a collection of possible outcomes. The set of events is a collection of such subsets, including the sample space itself and the empty set. In formal terms, this collection is required to be a σ-algebra (sigma-algebra), which means it is closed under complementation and countable unions. For example, the event "rolling an even number" can be represented as the set ${2, 4, 6}$.

3. **Probability Measure ($\mathcal{P}$)**: This is a function that assigns a probability to each event in the set of events. The probability measure has to satisfy three axioms:
- Non-negativity: For every event A in $\mathcal{F}$, $\mathcal{P}(A) ≥ 0$.
- Normalization: $\mathcal{P}(\Omega) = 1$, meaning the probability of the entire sample space (i.e., something happens) is $1$.
- Additivity: For any countable sequence of mutually exclusive events (events that cannot happen simultaneously), the probability of the union of these events is equal to the sum of their individual probabilities.

The probability space provides a rigorous mathematical foundation for probability theory, allowing for the analysis and modeling of random processes and the computation of probabilities in a consistent and well-defined manner.