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$E$ is Lindelöf, as a separable (second countable) metric space. So we have a cover of $E$ by these $(x-r_x,x+r_x)$ and so a countable subcover $\{(x-r_x,x+r_x)\cap E: x \in N \}$ exists for some countable subset $N$ of $E$. But then (as we have a cover of $E$): $$\lambda(E) \le \sum_{x \in N} \lambda((x-r_x,x+r_x) \cap E) = 0$$ by countable subadditivity ...

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The basic definition is that in a topological space $\Omega$, an $F_{\sigma}$ set is a countable union of closed sets and a $G_{\delta}$ set is the complement of an $F_{\sigma}$ set, i.e. a countable intersection of closed sets (using De Morgan). I have only encountered $F_{\sigma}$ and $G_\delta$ sets in Measure Theory, where they play a natural role, ...

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The probability that $$X_n(\omega)=\omega^n\leq x$$ is the Lebesgue measure of the set $$\{\omega \in [0,1] | \omega^n\leq x \}=\{\omega\in [0,1] |\omega\leq x^{\frac 1 n}\}$$ which is $$x^{\frac 1 n}, x\in [0,1],$$ and zero below $0$, and $1$ above $1$. The derivative of the function above is the density of $X_n$. Your mistakes are shown in red below: ...

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I'm not sure if this provides a complete (heh) answer, but here goes: For the first question in your final paragraph, it's exactly what it's defined to be; I'm not sure what else to really say about that. For your second question in the final paragraph, it depends. If $X$ and $Y$ are $\sigma$-compact metric spaces, then $\mathcal{B}_{X\times Y}=\mathcal{B}... 2 For any non-negative number$x$we have$x^{p} \leq x^{r}+x^{q}$. In fact$x^{p} \leq x^{r}$if$x \leq 1$and$x^{p} \leq x^{q}$if$x >1$. Hence$|f|^{p} \leq |f|^{r}+|f|^{q}$. Integrate both sides to derive the stated inequality. 1$\int |f|^{r} \leq (\int |f|^{p})^{r/p}(\mu(X))^{1-r/p}$so$\|f||_r \leq\|f\|_p (\mu(X))^{s}$. I have applied Holder's inequality with indices$\frac p r$and$\frac p {p-r}$. 1 Bruckner, Bruckner and Thompson's Real Analysis is a rather good book for these kinds of sets: they introduce them in the first chapter (starting at page 5!) and provide a good breakdown of how they develop$F_{\sigma \delta}, F_{\sigma \delta \sigma}, \ldots$and the inclusions, and they provide structured exercises to further improve your understanding. ... 1 Assume the closure of a set of points$\mathcal{P}$has$0$Lebesgue measure. For any interval$[a,b] \subset \mathbb{R}$and any$\delta > 0$, does there exist a finite number of open intervals whose union has measure$\delta$that covers$\mathcal{P} \cap [a,b]$? Yes. If$\overline{P}$has measure zero, then$\overline{P} \cap [a,b]\$ is a compact set ...

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