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Math4121 Exam 2 Review

Range: Chapter 2-4 of Bressoud's A Radical Approach to Lebesgue's Theory of Integration

Chapter 2

The Riemann-Stieltjes Integral

Definition of the Riemann-Stieltjes Integral

Let f be a bounded function on [a,b] and \alpha be a bounded function on [a,b].

We say that f is Riemann-Stieltjes integrable with respect to \alpha on [a,b] if there exists a number I such that for every \epsilon > 0, there exists a \delta > 0 such that for every partition P = \{a = x_0, x_1, \ldots, x_n = b\} of [a,b] with ||P|| < \delta, we have

\left| \int_a^b f \, d\alpha - I \right| < \epsilon

If f is Riemann-Stieltjes integrable with respect to \alpha on [a,b], we write

\int_a^b f \, d\alpha = I

Darboux Sums

Let P = \{a = x_0, x_1, \ldots, x_n = b\} be a partition of [a,b].

The upper Darboux sum of f with respect to \alpha is

U(f, \alpha, P) = \sum_{i=1}^n M_i (x_i - x_{i-1})

where M_i = \sup_{x \in [x_{i-1}, x_i]} f(x) and \alpha_i = \sup_{x \in [x_{i-1}, x_i]} \alpha(x).

The lower Darboux sum of f with respect to \alpha is

L(f, \alpha, P) = \sum_{i=1}^n m_i (x_i - x_{i-1})

where m_i = \inf_{x \in [x_{i-1}, x_i]} f(x) and \alpha_i = \inf_{x \in [x_{i-1}, x_i]} \alpha(x).

Fail of Riemann-Stieltjes Integration

Consider the function

((x)) = \begin{cases} 
x-\lfloor x \rfloor & x \in [\lfloor x \rfloor, \lfloor x \rfloor + \frac{1}{2}) \\
0 & x=\lfloor x \rfloor + \frac{1}{2}\\
x-\lfloor x \rfloor - 1 & x \in (\lfloor x \rfloor + \frac{1}{2}, \lfloor x \rfloor + 1] \end{cases}

We define

f(x) = \sum_{n=1}^{\infty} \frac{((nx))}{n^2}=\lim_{N\to\infty}\sum_{n=1}^{N} \frac{((nx))}{n^2}

(i) The series converges uniformly over x\in[0,1].

\left|f(x)-\sum_{n=1}^{N} \frac{((nx))}{n^2}\right|\leq \sum_{n=N+1}^{\infty}\frac{|((nx))|}{n^2}\leq \sum_{n=N+1}^{\infty} \frac{1}{n^2}<\epsilon

As a consequence, f(x)\in \mathscr{R}.

(ii) f has a discontinuity at every rational number with even denominator.

\begin{aligned}
\lim_{h\to 0^+}f(\frac{a}{2b}+h)-f(\frac{a}{2b})&=\lim_{h\to 0^+}\sum_{n=1}^{\infty}\frac{((\frac{na}{2b}+h))}{n^2}-\sum_{n=1}^{\infty}\frac{((\frac{na}{2b}))}{n^2}\\
&=\lim_{h\to 0^+}\sum_{n=1}^{\infty}\frac{((\frac{na}{2b}+h))-((\frac{na}{2b}))}{n^2}\\
&=\sum_{n=1}^{\infty}\lim_{h\to 0^+}\frac{((\frac{na}{2b}+h))-((\frac{na}{2b}))}{n^2}\\
&>0
\end{aligned}

Some integrable functions are not differentiable (violates the fundamental theorem of calculus)

Solve:

Define the oscilation of f on [x_{i-1}, x_i] as

\omega(f, [x_{i-1}, x_i]) = \sup_{x,y \in [x_{i-1}, x_i]} |f(x) - f(y)|-\inf_{x,y \in [x_{i-1}, x_i]} |f(x) - f(y)|

And define continuous functions as those functions that have oscilation 0 on every subinterval of their domain.

that is, the function f is continuous at c if \omega(f,c) = 0.

And we claim that the function is integrable on [a,b] if and only if the outer measure of the set of discontinuities of f is 0.

Finite cover:

Given a set S, an finite cover of S is a finite collection of open/ or closed/ or half-open intervals \{I_1, I_2, \ldots, I_n\} such that S \subseteq \bigcup_{i=1}^n I_i. The set of all finite covers of S is denoted by \mathcal{C}_S.

Length of a cover:

The length of a cover \ell(C) is the sum of the lengths of the intervals in the cover. (open/closed/half-open doesn't matter.)

Outer content:

The outer content of a set S is the infimum of the lengths of all finite covers of S. c_e(S) = \inf_{C\in \mathcal{C}_S}\ell(C). (e denotes "exterior")

Homework question: You cannot cover an interval [a,b] with length k with a finite cover of length strictly less than k.

Proceed by counting the intervals I_i = [l_i, r_i] in the cover, and r_n-l_0 is less than or equal to c_e(S) and l_0\leq a and r_n\leq b.

Theorem 2.5

Given a bounded function f defined on the interval [a,b], let S_\sigma be the points in [a,b] with oscilation greater than \sigma.

The function f is Riemann-Stieltjes integrable over [a,b] if and only if \lim_{\sigma \to 0} |S_\sigma| = 0. That is, for every \sigma > 0, the outer content of S_\sigma is 0.

Extra terminology:

Dense:

A set S is dense in the interval I is every open subinterval of I contains a point of S.

This is equivalent to saying that S is dense in I if every point of I is a limit point of S or a point of S. (proved in homework)

Totally discontinuous:

A discontinuous function is totally discontinuous in an interval if the set of points of continuity is not dense in that interval.

In other words, there exists an open interval I such that the set of points of continuity of f in I is empty.

Pointwise discontinuity:

A discontinuous function is pointwise discontinuous if the set of points of discontinuity is dense in the domain of f.

Accumulation point (limit point):

A point p is an accumulation point of a set S if every neighborhood of p contains a point of S other than p itself. (That is, there exists a convergent sequence \{p_n\}_{n=1}^\infty in S such that \lim_{n\to\infty} p_n = p and p_n \neq p for all n \in \mathbb{N}. Proved in Rudin)

Derived set:

The derived set of a set S is the set of all accumulation points of S. S' = \{p \in \mathbb{R} \mid \forall \epsilon > 0, \exists x \in S \text{ s.t. } 0 < |x-p| < \epsilon\}.

Type 1 set:

A set S is a type 1 set if S'\neq \emptyset and S''=\emptyset.

Type n set:

A set S is a type n set if S' is a type n-1 set.

First species:

A set S is of first species if it is type n for some n\geq 0, otherwise it is of second species.

\mathbb{Q} is not first species since it is dense in \mathbb{R} and \mathbb{Q}' = \mathbb{R}.

\mathbb{R} is not first species.

Chapter 3

Topology of \mathbb{R}

Open set:

A set S is open if for every x \in S, there exists an \epsilon > 0 such that B_\epsilon(x) \subseteq S.

Closed set:

A set S is closed if its complement is open.

Equivalently, a set S is closed if it contains all of its limit points. That is S' \subseteq S.

Interior of a set:

The interior of a set S is the set of all points in S such that there exists an \epsilon > 0 such that B_\epsilon(x) \subseteq S. S^\circ = \{x \in S \mid \exists \epsilon > 0 \text{ s.t. } B_\epsilon(x) \subseteq S\}. (It is also the union of all open sets contained in S.)

Closure of a set:

The closure of a set S is the set of all points that for every \epsilon > 0, B_\epsilon(x) \cap S \neq \emptyset. \overline{S} = \{x \in \mathbb{R} \mid \forall \epsilon > 0, B_\epsilon(x) \cap S \neq \emptyset\}.

Boundary of a set:

The boundary of a set S is the set of all points in S that are not in the interior of S. \partial S = \overline{S} \setminus S^\circ.

Theorem 3.4

Bolzano-Weierstrass Theorem:

Every bounded infinite set has an accumulation point.

Proof:

Let S be a bounded infinite set. Cut the interval [a,b] into two halves, and let I_1 be one with infinitely many points of S. (such set exists since S is infinite.)

Let I_2 be the one half with infinitely many points of I_1.

By induction, we can cut the interval into two halves, and let I_{n+1} be the one half with infinitely many points of I_n.

By the nested interval property, there exists a point c that is in all I_n.

c is an accumulation point of S.

QED

Theorem 3.6 (Heine-Borel Theorem)

For any open cover of a compact set, there exists a finite subcover.

Compact set:

A set S is compact if every open cover of S has a finite subcover. In \mathbb{R}, this is equivalent to being closed and bounded.

Cardinality:

The cardinality of \mathbb{R} is \mathfrak{c}.

The cardinality of \mathbb{N}, \mathbb{Z}, and \mathbb{Q} is \aleph_0.

Chapter 4

Nowhere Dense set

A set S is nowhere dense if there are no open intervals in which S is dense.

That is equivalent to S' contains no open intervals.

Note: If S is nowhere dense, then S^c is dense. But if S is dense, S^c is not necessarily nowhere dense. (Consider \mathbb{Q})

Perfect Set

A set S is perfect if S'=S.

Example: open intervals, Cantor set.

Cantor set

The Cantor set (SVC(3)) is the set of all real numbers in [0,1] that can be represented in base 3 using only the digits 0 and 2.

The outer content of the Cantor set is 0.

Generalized Cantor set (SVC(n))

The outer content of SVC(n) is \frac{n-3}{n-2}.

Lemma 4.4

Osgood's Lemma:

Let G be a closed, bounded set and Let G_1\subseteq G_2\subseteq \ldots and G=\bigcup_{n=1}^{\infty} G_n. Then \lim_{n\to\infty} c_e(G_n)=c_e(G).

Key: Using Heine-Borel Theorem.

Theorem 4.5

Arzela-Osgood Theorem:

Let \{f_n\}_{n=1}^{\infty} be a sequence of continuous, uniformly bounded functions on [0,1] that converges pointwise to 0. It follows that

\lim_{n\to\infty}\int_0^1 f_n(x) \, dx = \int_0^1 \lim_{n\to\infty} f_n(x) \, dx=0

Key: Using Osgood's Lemma and do case analysis on bounded and unbounded parts of the Riemann-Stieltjes integral.

Theorem 4.7

Baire Category Theorem:

An open interval cannot be covered by a countable union of nowhere dense sets.