数学代写|MATH5011 Real Analysis 👍 👍 - 👍代考代写：100%准时可靠您的作业代写专家

The mid-term exam of Math 5011 will be held at LSB, LT6 next Friday, Oct 21, 2022, 18:30 pm-21:30pm. There will be no lecture in the morning of next Friday (Oct 21).

Course Description
Abstract integration theory; outer measures and Caratheodory’s construction, Borel, Radon and Hausdorff measures; positive linear functionals and Riesz representation theorem; Lp-spaces and their functional properties; and signed measures, Radon-Nikodym theorem and the dual of the space of continuous functions.

MATH5011 Real Analysis HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

Let $B$ be the set defined in (1). Let $\mu$ be a measure on $(X, \mathcal{M})$. Show that
$$
\mu(B) \leq \liminf _{k \rightarrow \infty} \mu\left(A_k\right) .
$$

Solution Using the characterization $$ B=\bigcup{k=1}^{\infty} \bigcap_{j \geq k} A_j,
$$
and the fact that $\left{\cap_{j \geq k} A_j\right}$ is ascending in $k$, we have
$$
\begin{aligned}
\mu(B) & =\lim {k \rightarrow \infty} \mu\left(\bigcap{j \geq k} A_j\right) \
& =\liminf {k \rightarrow \infty} \mu\left(\bigcap{j \geq k} A_j\right) \
& \leq \liminf _{k \rightarrow \infty} \mu\left(A_k\right)
\end{aligned}
$$

问题 2.

Here we review Riemann integral. Let $f$ be a bounded function defined on $[a, b], a, b \in \mathbb{R}$. Given any partition $P=\left{a=x_0<x_1<\cdots<x_n=b\right}$ on $[a, b]$ and tags $z_j \in\left[x_j, x_{j+1}\right]$, there corresponds a Riemann sum of $f$ given by $R(f, P, \mathbf{z})=\sum_{j=0}^{n-1} f\left(z_j\right)\left(x_{j+1}-x_j\right)$. The function $f$ is called Riemann integrable with integral $L$ if for every $\varepsilon>0$ there exists some $\delta$ such that
$$
|R(f, P, \mathbf{z})-L|<\varepsilon
$$
whenever $|P|<\delta$ and $\mathbf{z}$ is any tag on $P$. (Here $|P|=\max {j=0}^{n-1}\left|x{j+1}-x_j\right|$ is the length of the partition.) Show that

For any partition $P$, define its Darboux upper and lower sums by
$$
\bar{R}(f, P)=\sum_j \sup \left{f(x): x \in\left[x_j, x_{j+1}\right]\right}\left(x_{j+1}-x_j\right)
$$
and
$$
R(f, P)=\sum_j \inf \left{f(x): x \in\left[x_j, x_{j+1}\right]\right}\left(x_{j+1}-x_j\right)
$$
respectively. Show that for any sequence of partitions $\left{P_n\right}$ satisfying $\left|P_n\right| \rightarrow 0$ as $n \rightarrow \infty, \lim {n \rightarrow \infty} \bar{R}\left(f, P_n\right)$ and $\lim {n \rightarrow \infty} R\left(f, P_n\right)$ exist.
$\left{P_n\right}$ as above. Show that $f$ is Riemann integrable if and only if
$$
\lim {n \rightarrow \infty} \bar{R}\left(f, P_n\right)=\lim {n \rightarrow \infty} R\left(f, P_n\right)=L
$$
A set $E$ in $[a, b]$ is called of measure zero if for every $\varepsilon>0$, there exists a countable subintervals $J_n$ satisfying $\sum_n\left|J_n\right|<\varepsilon$ such that $E \subset \bigcup_n J_n$. Prove Lebsegue’s theorem which asserts that $f$ is Riemann integrable if and only if the set consisting of all discontinuity points of $f$ is a set of measure zero. Google for help if necessary.

Solution:
(a) It suffices to show: For every $\varepsilon>0$, there exists some $\delta$ such that
$$
0 \leq \bar{R}(f, P)-\bar{R}(f)<\varepsilon
$$

and
$$
0 \leq R(f)-R(f, P)<\varepsilon $$ for any partition $P,|P|<\delta$, where $$ \bar{R}(f)=\inf P \bar{R}(f, P) $$ and $$ R(f)=\sup _P R(f, P) $$ If it is true, then $\lim {n \rightarrow \infty} \bar{R}\left(f, P_n\right)$ and $\lim _{n \rightarrow \infty} R\left(f, P_n\right)$ exist and equal to $\bar{R}(f)$ and $R(f)$ respectively. Given $\varepsilon>0$, there exists a partition $Q$ such that
$$
\bar{R}(f)+\varepsilon / 2>\bar{R}(f, Q)
$$

Let $m$ be the number of partition points of $Q$ (excluding the endpoints). Consider any partition $P$ and let $R$ be the partition by putting together $P$ and $Q$. Note that the number of subintervals in $P$ which contain some partition points of $Q$ in its interior must be less than or equal to $m$. Denote the indices of the collection of these subintervals in $P$ by $J$. We have
$$
0 \leq \bar{R}(f, P)-\bar{R}(f, R) \leq \sum_{j \in J} 2 M \Delta x_j \leq 2 M \times m|P|,
$$
where $M=\sup _{[a, b]}|f|$, because the contributions of $\bar{R}(f, P)$ and $\bar{R}(f, Q)$ from the subintervals not in $J$ cancel out. Hence, by the fact that $R$ is a refinement of $\mathrm{Q}$,
$$
\bar{R}(f)+\varepsilon / 2>\bar{R}(f, Q) \geq \bar{R}(f, R) \geq \bar{R}(f, P)-2 M m|P|,
$$

i.e.,
$$
0 \leq \bar{R}(f, P)-\bar{R}(f)<\varepsilon / 2+2 M m|P|
$$
Now, we choose
$$
\delta<\frac{\varepsilon}{1+4 M m}
$$
Then for $P,|P|<\delta$,
$$
0 \leq \bar{R}(f, P)-\bar{R}(f)<\varepsilon
$$
Similarly, one can prove the second inequality.

问题 3.

Let $g$ be a measurable function in $[0, \infty]$. Show that
$$
m(E)=\int_E g d \mu
$$
defines a measure on $\mathcal{M}$. Moreover,
$$
\int_X f d m=\int_X f g d \mu, \quad \forall f \text { measurable in }[0, \infty] .
$$

Solution: We readily check that
(1) $m(\phi)=0$
(2) $m(E) \geq 0, \forall E \in M$
(3) For mutually disjoint $A_k \in M$,
$$
m\left(\bigcup_{k=1}^{\infty} A_k\right)=\int_X \sum_{k=1}^{\infty} \chi_{A_k} g d \mu=\sum_{k=1}^{\infty} \int \chi_{A_k} g d \mu=\sum_{k=1}^{\infty} m\left(A_k\right)
$$
by monotone convergence theorem, since $\sum_{k=1}^n \chi_{A_k} g \uparrow \sum_{k=1}^{\infty} \chi_{A_k} g$.
To prove the last assertion, consider the following cases:
(a) $f=\chi_E$ for some $E \in M$.
$$
\int_X f d m=\int_E d m=m(E)=\int_E g d \mu=\int_X \chi_E g d \mu=\int_X f g d \mu .
$$

(b) $f$ is a non-negative simple function.
This follows from (a).
(c) $f$ is a non-negative measurable function.
Pick a sequence $s_n \geq 0$ of simple functions such that $s_n \uparrow f$ pointwisely.
Then $0 \leq s_n g \uparrow g$ pointwisely. From (b),
$$
\int_X s_n d m=\int_X s_n g d \mu
$$
Taking $n \rightarrow \infty$, by monotone convergence theorem, we have
$$
\int_X f d m=\int_X f g d \mu .
$$

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