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1) Let $\Omega=[0,1]$, $F = B([0,1])$, $P$ be Lebesgue measure on $[0,1]$ ($P([a,b])=b-a$). Set $$A_n^i:=\left[{\frac{i-1}{n},\frac{i}{n}}\right]$$ and $$X_n^i(\omega):=\chi_{A_n^i}(\omega)$$

Consider the sequences $$\{X_1^1,X_2^1,X_2^2,X_3^1,X_3^2,X_3^3\},\quad n \geq 1,\,i=1,2,\dots,n$$

Show that this sequence converges to the zero r.v. both in probability and in $L^2(\Omega, F, P)$, but does not converge at any point $\omega \in [0,1]$.

2) Let the probability space $(\Omega, F, P)$ be as above. Find a sequence of random variables $\{X_n\}$ that converges to the zero random variable a.s. but such that $E|X_n|^p \rightarrow \infty$ for every $0 < p < \infty$.

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closed as off-topic by Davide Giraudo, Math1000, mathematics2x2life, José Carlos Santos, JonMark Perry May 3 '18 at 10:17

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