Kuba Helsztyński
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 6 Equivalent metrics determine the same topology 5 Counterexample for $(p\rightarrow q) \longleftrightarrow (!q \rightarrow\mathord !p)$ 4 Show $\lim\limits_{n\to\infty} \sqrt[n]{n^e+e^n}=e$ 3 $\mu(E\setminus (E+x))=0$ for all $x\in\mathbb{R}$. Prove that $\mu(E)=0$ or $\mu(\mathbb{R}\setminus E)=0$ 2 Why is the probability that $(X_1+\ldots+X_n)/n$ converges either $0$ or $1$?

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### Questions (13)

 6 $\mu(E\setminus (E+x))=0$ for all $x\in\mathbb{R}$. Prove that $\mu(E)=0$ or $\mu(\mathbb{R}\setminus E)=0$ 4 Prove that $\frac{1}{2h}\int_a^b\mu(A\cap(x-h,x+h))\,\text{d}x\le \mu(A)$ 4 Do probability measures have to be the same if they agree on a generator of Borel $\sigma$–algebra $\mathcal{B}(\mathbb{R})$? 3 Favourable modification of “Double or Nothing” 3 How to prove that $E(X\mid\sigma\,(\mathcal{G},\mathcal{M}))=E(X\mid\mathcal{M})$ if $\mathcal{G}$ is independent of $X$ and $\mathcal{M}$?

### Tags (27)

 8 metric-spaces × 2 4 probability-theory × 8 7 general-topology × 2 4 limits 6 logic × 2 3 probability × 4 5 measure-theory × 10 2 stopping-times × 2 5 propositional-calculus 2 random-variables × 2

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