Measure-theoretic probability

Let $X, X_1, X_2, \dots$ be random variables on one probability space. Which implications hold in general? Select all that apply.

Let $X_1, X_2$ be independent fair dice (values $1, \dots, 6$). The conditional expectation $\mathbb E[X_1 X_2 \mid X_1]$ is a random variable. What value does it take on the event $\{X_1 = 4\}$?

Let $A_1, A_2, \dots$ be events with $P(A_n) = 1/n^2$. Nothing is assumed about independence. Which statement is true?

The proof of the second Borel–Cantelli lemma runs: (1) the complement of $\limsup A_k$ is $\bigcup_n \bigcap_{k \ge n} A_k^c$; (2) $P(\bigcap_{k=n}^N A_k^c) = \prod_{k=n}^N (1 - P(A_k))$; (3) $1 - x \le e^{-x}$; (4) continuity from above. Which step uses the independence of the events?

On $\Omega = [0, 1)$ with Lebesgue measure let $X(\omega) = \omega$ and let $\mathcal G$ be the σ-algebra generated by $B_1 = [0, \tfrac12)$ and $B_2 = [\tfrac12, 1)$. What is $\mathbb E[X \mid \mathcal G]$?

In Mathlib, Martingale M 𝓕 μ means: M is adapted and μ[M j | 𝓕 i] =ᵐ[μ] M i whenever i ≤ j. Which Lean statement correctly expresses a martingale has constant expectation (with [IsFiniteMeasure μ] and hM : Martingale M 𝓕 μ, hij : i ≤ j)?

Let $X_1, \dots, X_4$ be independent fair signs, $P(X_i = 1) = P(X_i = -1) = \tfrac12$, and $S_4 = X_1 + X_2 + X_3 + X_4$. Compute $\mathbb E[S_4^4]$ using the expansion from the proof of the strong law.

A simple symmetric random walk starts at $3$ and stops at the first time $T$ it reaches $0$ or $10$. Using optional stopping for the martingales $S_n$ and $S_n^2 - n$, compute $\mathbb E T$.

Let $\xi_1, \xi_2, \dots$ be independent with $P(\xi_i = 0) = P(\xi_i = 2) = \tfrac12$ and $M_n = \xi_1 \cdots \xi_n$. Then $(M_n)$ is a non-negative martingale with $\mathbb E M_n = 1$, and $M_n \to 0$ almost surely. Why does this not contradict $\mathbb E M_\infty = \lim \mathbb E M_n$?

A finite probability space has outcomes $0, 1, \dots, n-1$ with probabilities probs, a random variable given by the list X, and a σ-algebra $\mathcal G$ generated by a partition blocks (a list of lists of outcomes). Write conditional_expectation(probs, X, blocks) returning the list whose $\omega$-th entry is $\mathbb E[X \mid \mathcal G](\omega)$. Assume every block has positive probability. Use Fraction so the answers are exact.