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A probability measure is a function that assigns a probability to each event in a sample space. It must satisfy certain properties such as non-negativity, normalization, and additivity. In simpler terms, a probability measure helps us calculate how likely an event is to occur.
For any event S within a sample space \(\Omega\), the probability measure \(\mathrm{P}(S)\) should provide numbers between 0 and 1. These numbers express the likelihood of S happening. Here's how it breaks down:

• Non-Negativity: Probabilities cannot be negative.
• Normalization: The total probability of all possible events is 1.
• Additivity: When considering multiple disjoint events, the total probability is the sum of their individual probabilities.

Understanding the properties of a probability measure is crucial in studying conditional probability. Essentially, it forms the foundation of how probabilities are calculated and interpreted.

Normalization ensures that the total probability sums to 1. This is essential because it implies that something within the sample space is guaranteed to happen. In the context of conditional probability, normalization plays a specific role.
When we have an event H with \(\mathrm{P}(H)>0\), the conditional probability \(\mathrm{P}(S \mid H)\) is defined as:
\( \mathrm{P}(S \mid H) = \frac{ \mathrm{P}(S \cap H) }{ \mathrm{P}(H) } \)
For normalization, we need to show that the sum of conditional probabilities over the entire sample space equals 1. This can be demonstrated as:
\(\sum_{S \in \Omega} \mathrm{P}(S \mid H) = \sum_{S \in \Omega} \frac{ \mathrm{P}(S \cap H) }{ \mathrm{P}(H) } = \frac{ 1 }{ \mathrm{P}(H) } \sum_{ S \in \Omega } \mathrm{P}(S \cap H)\)
This matches with the fact that \(\textbf{S} \cap \textbf{H} \) partition \(\textbf{H}\), leading to \(\sum_{S \in \Omega} \mathrm{P}(S \mid H) = 1 \)
This assures us that considering all potential outcomes, something is certain to occur.

Non-negativity means that the probability of any event is never less than zero. For any event S, \(\mathrm{P}(S \mid H) \) should be non-negative:
\( \mathrm{P}(S \mid H) = \frac{ \mathrm{P}(S \cap H) }{ \mathrm{P}(H) } \)
Both the numerator \(\mathrm{P}(S \cap H)\) and the denominator \(\mathrm{P}(H)\) here are non-negative because they represent probabilities. Probabilities, by definition, cannot be negative.
This means we are dividing a non-negative value by another non-negative value, ensuring that:
\( \mathrm{P}(S \mid H) \geq 0 \)
Non-negativity is essential for the integrity of probability measures, reinforcing that probabilities always fall within the 0 to 1 range.

Additivity helps us figure out the probability of multiple disjoint events happening. If we have disjoint events \(S_1, S_2, S_3, ...\) that cannot occur simultaneously, the probability of their union is the sum of their individual probabilities:
For conditional probability, this property looks like:
\( \mathrm{P}((S_1 \cup S_2 \cup ... ) \mid H) = \frac{ \mathrm{P}((S_1 \cup S_2 \cup ... ) \cap H) }{ \mathrm{P}(H) } \)
Since the events \(\textbf{S}_i \) are disjoint, it follows:
\( \mathrm{P}((S_1 \cup S_2 \cup ... ) \mid H) = \frac{ \mathrm{P}(S_1 \cap H) + \mathrm{P}(S_2 \cap H) + ... }{ \mathrm{P}(H) } \)
This equals:
\( \mathrm{P}(S_1 \mid H) + \mathrm{P}(S_2 \mid H) + ...\)
This property is known as countable additivity and is a fundamental aspect of probability measures. It simplifies working with probabilities of multiple events occurring.