5. Let (Ω, ℱ, ℙ) be a probability space, and A1, A2, … be an increasing sequence of events; that

Step 1: We need to show that the sequence of events converges in probability to the event A = U∞

Question

5. Let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, and $A_1, A_2, \dots$ be an increasing sequence of events; that is, $A_1 \subseteq A_2 \subseteq \dots$ such that $\mathbb{P}(A_1) = \mathbb{P}(A_2) = \dots = p > 0$. Does the sequence of events converge in probability to the event $A = \bigcup_{n=1}^{\infty} A_n$? Prove or disprove this. [2] Hint: For a sequence of events, convergence in probability can be written as $\lim_{n \to \infty} \mathbb{P}(A \triangle A_n) = 0$, where $\triangle$ is the symmetric difference.

          5. Let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, and $A_1, A_2, \dots$ be an increasing sequence of events; that is, $A_1 \subseteq A_2 \subseteq \dots$ such that $\mathbb{P}(A_1) = \mathbb{P}(A_2) = \dots = p > 0$. Does the sequence of events converge in probability to the event $A = \bigcup_{n=1}^{\infty} A_n$? Prove or disprove this. [2]
Hint: For a sequence of events, convergence in probability can be written as $\lim_{n \to \infty} \mathbb{P}(A \triangle A_n) = 0$, where $\triangle$ is the symmetric difference.

5. Let (Ω, ℱ, ℙ) be a probability space, and A1, A2, … be an increasing sequence of events; that is, A1 ⊆ A2 ⊆… such that ℙ(A1) = ℙ(A2) = … = p > 0. Does the sequence of events converge in probability to the event A = ⋃n=1^∞ An? Prove or disprove this. [2]
Hint: For a sequence of events, convergence in probability can be written as limn →∞ℙ(A An) = 0, where is the symmetric difference.

Added by Kevin N.

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