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3

$E$ 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 ...


2

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, ...


2

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: ...


2

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 $\frac p {p-r}$.


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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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