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MATH5011课程简介
Announcement
- Here is the course outline [Download file]
- 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.
Prerequisites
- Real and Complex Analysis, 3rd ed. W. Rudin, McGrawHill, New York 1966.
- Measure Theory and Fine Properties of Functions, L.C. Evans and R.F. Gariepy, CRC Press 1992.
- Real Analysis: Measure Theory, Integration and Hilbert Spaces, E.M. Stein and R. Shakarchi, Princeton Lectures in Analysis, Princeton 2005.
- Real and Abstract Analysis, E. Hewitt and K. Stromberg, Graduate Texts in Mathematics, Springer-Verlag, New York 1975.
MATH5011 Real Analysis HELP（EXAM HELP， ONLINE TUTOR）
(1) Let $\left{A_k\right}_{k=1}^{\infty}$ be a sequence of measurable sets in $(X, \mathcal{M})$. Let
$$
A=\left{x \in X: x \in A_k \text { for infinitely many } \mathrm{k}\right},
$$
and
$$
B=\left{x \in X: x \in A_k \text { for all except finitely many k }\right} \text {. }
$$
Show that $A$ and $B$ are measurable.
$\begin{aligned} & A=\bigcap_{n=1}^{\infty} \bigcup_{k \geq n} A_k . \ & B=\bigcup_{n=1}^{\infty} \bigcap_{k \geq n} A_k .\end{aligned}$
(2) Let $\Psi: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Show that $\Psi(f, g)$ are measurable for any measurable functions $f, g$. This result contains Proposition 1.3 as a special case.
Solution Note that every open set $G \subseteq \mathbb{R}^2$ can be written as a countable union of set of the form $V_1 \times V_2$ where $V_1, V_2$ open in $\mathbb{R}$. (Think of $V_1 \times V_2=(a, b) \times$ $(c, d), a, b, c, d \in Q)$.
Let $G \subseteq \mathbb{R}$ be open. Then $\Phi^{-1}(G)$ is open in $\mathbb{R}^2$, so
$$
\Phi^{-1}(G)=\bigcup_n\left(V_n^1 \times V_n^2\right)
$$
Then
$$
h^{-1}\left(\Phi^{-1}\right)(G)=\bigcup_n h^{-1}\left(V_n^1 \times V_n^2\right)=\bigcup_n f^{-1}\left(V_n^1\right) \cap g^{-1}\left(V_n^2\right)
$$
is measurable since $f$ and $g$ are measurable. Hence $h=(f, g)$.
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