Kuba Helsztyński
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1,027
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 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 $\mu(\mathbb{R}\setminus E)=0$ 3 Equivalent metrics determine the same topology 2 Is a linear map of norm $1$ always an isometry?

### Reputation (1,027)

 +5 How to prove independence of applicants' relative ranks in secretary problem? +5 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 $\mathcal{M}$? +5 Is it true that $\mathbb{E}(X\mid X\leq x)\leq \mathbb{E}(X\mid A)$ whenever $\mathbb{P}(X\leq x)\leq \mathbb{P}(A)$? +5 $\mu(E\setminus (E+x))=0$ for all $x\in\mathbb{R}$. Prove that $\mu(E)=0$ or $\mu(\mathbb{R}\setminus E)=0$

### 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)

 6 logic × 2 4 limits 5 measure-theory × 10 3 probability × 4 5 metric-spaces × 2 2 probability-theory × 8 5 propositional-calculus 2 real-analysis × 2 4 general-topology × 2 2 normed-spaces

### Accounts (2)

 Mathematics 1,027 rep 1717 Academia 101 rep 1