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MATH5011课程简介
- 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.
Prerequisites
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.
MATH5011 Real Analysis HELP（EXAM HELP， ONLINE TUTOR）
Show that $f: X \rightarrow \overline{\mathbb{R}}$ is measurable if and only if $f^{-1}([a, b])$ is measurable for all $a, b \in \overline{\mathbb{R}}$.
Solution By def $f: X \rightarrow \bar{R}$ is measurable if $f^{-1}(G)$ is measurable. $\forall G$ open in $\bar{R}$. Every open set $G$ in $\bar{R}$ can be written as a countable union of $(a, b)$, $[-\infty, a),(b, \infty], a, b \in R$. So $f$ is measurable iff $f^{-1}(a, b), f^{-1}[-\infty, a), f^{-1}(b, \infty]$ are measurable.
$\Rightarrow)$ Use
$$
\begin{aligned}
f^{-1}(a, b) & =\bigcap_n f^{-1}\left(a-\frac{1}{n}, b+\frac{1}{n}\right) \
f^{-1}[-\infty, a) & =\bigcap_n f^{-1}\left[-\infty, a+\frac{1}{n}\right) \
f^{-1}(b, \infty] & =\bigcap_n f^{-1}\left(b-\frac{1}{n}, \infty\right]
\end{aligned}
$$
$\Leftarrow)$ Use
$$
\begin{aligned}
f^{-1}(a, b) & =\bigcup_n f^{-1}\left[a-\frac{1}{n}, b+\frac{1}{n}\right] \
f^{-1}[-\infty, a) & =\bigcap_n f^{-1}\left[-\infty, a-\frac{1}{n}\right] \
f^{-1}(b, \infty] & =\bigcap_n f^{-1}\left[b+\frac{1}{n}, \infty\right]
\end{aligned}
$$
Let $f: X \times[a, b] \rightarrow \mathbb{R}$ satisfy (a) for each $x, y \mapsto f(x, y)$ is Riemann integrable, and (b) for each $y, x \mapsto f(x, y)$ is measurable with respect to some $\sigma$-algebra $\mathcal{M}$ on $X$. Show that the function
$$
F(x)=\int_a^b f(x, y) d y
$$
is measurable with respect to $\mathcal{M}$.
Solution For simplicity let $[a, b]=[0,1]$. For $n \geq 1$, equally divide $[0,1]$ into subintervals of length $1 / n$ and let
$$
F_n(x)=\sum_{k=1}^n f\left(x, \frac{k}{n}\right) \frac{1}{n}
$$
Clearly $F_n$ is measurable (with respect to $\mathcal{M}$ ). Now
$$
F(x)=\lim _{n \rightarrow \infty} F_n(x)
$$
so it is also measurable.
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