. Using the properties of Definition 1.1.2 for a probability measure ℙ, show the following. (i)

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. Using the properties of Definition 1.1.2 for a probability...

. Using the properties of Definition 1.1.2 for a probability measure $\mathbb{P}$, show the following. (i) If $A \in \mathcal{F}, B \in \mathcal{F}$, and $A \subset B$, then $\mathbb{P}(A) \leq \mathbb{P}(B)$. (ii) If $A \in \mathcal{F}$ and $\left\{A_n\right\}_{n=1}^{\infty}$ is a sequence of sets in $\mathcal{F}$ with $\lim _{n \rightarrow \infty} \mathbb{P}\left(A_n\right)=0$ and $A \subset A_n$ for every $n$, then $\mathbb{P}(A)=0$. (This property was used implicitly in Example 1.1.4 when we argued that the sequence of all heads, and indeed any particular sequence, must have probability zero.)

. Using the properties of Definition 1.1.2 for a probability measure $\mathbb{P}$, show the following.
(i) If $A \in \mathcal{F}, B \in \mathcal{F}$, and $A \subset B$, then $\mathbb{P}(A) \leq \mathbb{P}(B)$.
(ii) If $A \in \mathcal{F}$ and $\left\{A_n\right\}_{n=1}^{\infty}$ is a sequence of sets in $\mathcal{F}$ with $\lim _{n \rightarrow \infty} \mathbb{P}\left(A_n\right)=0$ and $A \subset A_n$ for every $n$, then $\mathbb{P}(A)=0$. (This property was used implicitly in Example 1.1.4 when we argued that the sequence of all heads, and indeed any particular sequence, must have probability zero.)

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Key Concepts

Probability Measure

A probability measure is a function defined on a sigma algebra that assigns a non-negative real number to events, with the total measure of the sample space equal to one and countable additivity holding for disjoint events. This concept is foundational in probability theory, as it formalizes the idea of assigning probabilities to events in a mathematically rigorous way.

Monotonicity

Monotonicity is a key property of measures which states that if one event is a subset of another, then the measure (or probability) of the smaller event is less than or equal to that of the larger event. This is derived by expressing the larger event as the union of the smaller event and its complement relative to the larger set, and applying the additivity and non-negativity properties of the probability measure.

Continuity of Measure

Continuity of measure, particularly in the context of sequences of events, means that if you have a sequence of events whose probabilities approach zero and an event is contained in every one of these, then the probability of that event must also be zero. This property follows from monotonicity, ensuring that an event that is uniformly a subset of events with vanishing measure cannot have a positive probability.

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