Unit 7: Riemann & Riemann-Stieltjes Integration & Sequences of Functions

1. Partitions and Tagged Partitions

Let $[a, b] \subset \mathbb{R}$ be a compact interval.

Definition 7.1 (Partition and Mesh): A partition $P$ of $[a, b]$ is a finite ordered set of points:

The subintervals are $\Delta x_i = x_i - x_{i-1}$ for $i \in \{1, 2, \dots, n\}$. The mesh (or norm) of $P$ is the length of its longest subinterval:

A partition $P^$ is a refinement of $P$ if $P \subseteq P^$.

2. Darboux Upper and Lower Sums

Let $f: [a, b] \to \mathbb{R}$ be a bounded function. On each subinterval $[x_{i-1}, x_i]$, define:

Definition 7.2 (Darboux Sums):

1. The Upper Darboux Sum is:

1. The Lower Darboux Sum is:

1. Clearly, $L(f, P) \le U(f, P)$ for every partition $P$.

Lemma 7.1 (Refinement Lemma): If $P^$ is a refinement of $P$ ($P \subseteq P^$), then:

That is, refining a partition increases the lower sum and decreases the upper sum.

Proof Sketch:

If $P^*$ adds one point $c \in (x_{k-1}, x_k)$, the term $M_k \Delta x_k$ splits into $M_{k,1}(c - x_{k-1}) + M_{k,2}(x_k - c)$. Since $M_{k,1} \le M_k$ and $M_{k,2} \le M_k$, the sum can only decrease or stay equal. $\blacksquare$

Corollary 7.1: For any two arbitrary partitions $P_1$ and $P_2$:

Proof: Take the common refinement $P^ = P_1 \cup P_2$. Then $L(f, P_1) \le L(f, P^) \le U(f, P^*) \le U(f, P_2)$. $\blacksquare$

3. Darboux Upper and Lower Integrals & The Riemann Criterion

Definition 7.3 (Darboux Integrals):

1. The Upper Integral is:

1. The Lower Integral is:

By Corollary 7.1, $\underline{\int_a^b} f \le \overline{\int_a^b} f$ always.

Definition 7.4 (Riemann Integrability): A bounded function $f: [a, b] \to \mathbb{R}$ is Riemann integrable on $[a, b]$, denoted $f \in \mathcal{R}[a, b]$, if:

This common value is the Riemann integral, denoted $\int_a^b f(x) \, dx$.

Theorem 7.1 (Riemann / Darboux Integrability Criterion): A bounded function $f: [a, b] \to \mathbb{R}$ is Riemann integrable if and only if for every $\epsilon > 0$, there exists a partition $P_\epsilon$ of $[a, b]$ such that:

Proof:

$(\implies)$ If $\underline{\int} f = \overline{\int} f = I$, by definition of infimum and supremum, for any $\epsilon > 0$ there exist $P_1, P_2$ such that $U(f, P_1) < I + \epsilon/2$ and $L(f, P_2) > I - \epsilon/2$. Let $P_\epsilon = P_1 \cup P_2$. Then $U(f, P_\epsilon) - L(f, P_\epsilon) \le U(f, P_1) - L(f, P_2) < \epsilon$. $(\impliedby)$ For any partition $P$, $0 \le \overline{\int} f - \underline{\int} f \le U(f, P) - L(f, P) < \epsilon$. Since this holds for all $\epsilon > 0$, $\overline{\int} f = \underline{\int} f$. $\blacksquare$

1. Classes of Riemann Integrable Functions

Theorem 7.2 (Integrability of Continuous Functions): If $f: [a, b] \to \mathbb{R}$ is continuous on $[a, b]$, then $f \in \mathcal{R}[a, b]$.

Proof:

Since $[a, b]$ is compact and $f$ is continuous, by the Heine-Cantor Theorem (Theorem 5.7), $f$ is uniformly continuous on $[a, b]$. Let $\epsilon > 0$ be given. Choose $\delta > 0$ such that $|x - y| < \delta \implies |f(x) - f(y)| < \frac{\epsilon}{b - a}$. Choose any partition $P$ with mesh $\|P\| < \delta$. On each subinterval $[x_{i-1}, x_i]$, by the Extreme Value Theorem, $f$ attains its maximum at some $u_i$ and minimum at some $v_i$. Since $|u_i - v_i| \le \Delta x_i \le \|P\| < \delta$, we have:

Now evaluate the Darboux difference:

By the Riemann Criterion (Theorem 7.1), $f \in \mathcal{R}[a, b]$. $\blacksquare$

Theorem 7.3 (Integrability of Monotone Functions): If $f: [a, b] \to \mathbb{R}$ is monotonic on $[a, b]$, then $f \in \mathcal{R}[a, b]$. (Proof uses regular partition with equal subintervals $\Delta x = \frac{b - a}{n}$, leading to telescoping sum $(M_i - m_i)$).

2. Lebesgue's Criterion for Riemann Integrability

The definitive characterization of Riemann integrability was discovered by Henri Lebesgue using measure theory:

Definition 7.5 (Set of Measure Zero): A set $E \subset \mathbb{R}$ has Lebesgue measure zero if for every $\epsilon > 0$, there exists a countable collection of open intervals $\{ (a_k, b_k) \}_{k=1}^\infty$ covering $E$ such that:

Theorem 7.4 (Lebesgue-Vitali Integrability Theorem): A bounded function $f: [a, b] \to \mathbb{R}$ is Riemann integrable if and only if the set of its discontinuities has Lebesgue measure zero:

Implications:

• Any continuous function is integrable (discontinuity set is empty, measure 0).
• Any function with countably many discontinuities is integrable (every countable set has measure 0).
• Thomae's popcorn function is Riemann integrable on $[0, 1]$ (discontinuous only on $\mathbb{Q}$, which is countable).
• The Dirichlet indicator function $\mathbf{1}_\mathbb{Q}$ is discontinuous everywhere on $[0, 1]$. Its discontinuity set has measure 1, so it is not Riemann integrable.

1. Fundamental Theorem of Calculus (Part 1)

Theorem 7.5 (FTC Part 1 - Differentiation of the Integral): Let $f \in \mathcal{R}[a, b]$ and define the accumulation function:

1. $F$ is uniformly continuous (in fact, Lipschitz continuous) on $[a, b]$.
2. If $f$ is continuous at a point $x_0 \in (a, b)$, then $F$ is differentiable at $x_0$ and:

Complete Line-by-Line Proof:

Let $x_0 \in (a, b)$ be a point where $f$ is continuous. Consider the difference quotient of $F$ at $x_0$:

Let $\epsilon > 0$ be given. Since $f$ is continuous at $x_0$, there exists $\delta > 0$ such that:

For any $h$ with $0 < |h| < \delta$, all points $t$ between $x_0$ and $x_0 + h$ satisfy $|t - x_0| < \delta$. Therefore:

Since $\epsilon > 0$ was arbitrary, we conclude:

2. Fundamental Theorem of Calculus (Part 2)

Theorem 7.6 (FTC Part 2 - Evaluation Theorem): If $f: [a, b] \to \mathbb{R}$ is differentiable on $[a, b]$ and $f' \in \mathcal{R}[a, b]$, then:

Line-by-Line Proof:

Let $P = \{x_0, x_1, \dots, x_n\}$ be any partition of $[a, b]$. We express $f(b) - f(a)$ as a telescoping sum:

On each subinterval $[x_{i-1}, x_i]$, $f$ is continuous on $[x_{i-1}, x_i]$ and differentiable on $(x_{i-1}, x_i)$. By Lagrange's Mean Value Theorem (Theorem 6.6), there exists $c_i \in (x_{i-1}, x_i)$ such that:

Therefore:

Notice that on $[x_{i-1}, x_i]$, $m_i \le f'(c_i) \le M_i$, where $m_i = \inf_{[x_{i-1}, x_i]} f'$ and $M_i = \sup_{[x_{i-1}, x_i]} f'$. Multiplying by $\Delta x_i > 0$ and summing:

This inequality holds for every partition $P$ of $[a, b]$. Taking the supremum over all lower sums and infimum over all upper sums:

Since $f' \in \mathcal{R}[a, b]$, the upper and lower integrals coincide and equal $\int_a^b f'(t) \, dt$. Therefore:

3. The Riemann-Stieltjes Integral

Definition 7.6 (Riemann-Stieltjes Integral): Let $\alpha: [a, b] \to \mathbb{R}$ be a monotonically increasing integrator function. For partition $P$, define $\Delta \alpha_i = \alpha(x_i) - \alpha(x_{i-1}) \ge 0$. The Riemann-Stieltjes sums are formed by weighting by $\Delta \alpha_i$. If $\alpha$ is continuously differentiable ($\alpha \in C^1[a, b]$), the Stieltjes integral reduces to:

1. Pointwise vs. Uniform Convergence

Definition 7.7 (Pointwise Convergence): A sequence of functions $f_n: E \to \mathbb{R}$ converges pointwise to $f: E \to \mathbb{R}$ if for every $x \in E$:

That is, $\forall \epsilon > 0$ and $\forall x \in E$, $\exists N(\epsilon, x) \in \mathbb{N}$ such that $\forall n \ge N$, $|f_n(x) - f(x)| < \epsilon$.

Definition 7.8 (Uniform Convergence): A sequence of functions $f_n: E \to \mathbb{R}$ converges uniformly to $f$ on $E$, denoted $f_n \rightrightarrows f$, if for every $\epsilon > 0$, there exists an index $N \in \mathbb{N}$ (independent of $x$) such that:

Equivalently, in terms of the uniform norm (supremum norm):

2. The Uniform Cauchy Criterion

Theorem 7.7 (Uniform Cauchy Criterion): A sequence of functions $(f_n)$ converges uniformly on $E$ if and only if for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that:

3. The Weierstrass $M$-Test for Infinite Series of Functions

Theorem 7.8 (Weierstrass $M$-Test): Let $\sum_{n=1}^\infty f_n(x)$ be an infinite series of functions defined on $E \subseteq \mathbb{R}$. Suppose there exists a sequence of positive real constants $(M_n)_{n=1}^\infty$ such that:

1. $|f_n(x)| \le M_n$ for all $x \in E$ and all $n \in \mathbb{N}$.
2. The numerical series $\sum_{n=1}^\infty M_n$ converges.

Then the series of functions $\sum_{n=1}^\infty f_n(x)$ converges uniformly and absolutely on $E$.

Complete Proof:

Let $\epsilon > 0$. Since the scalar series $\sum M_n$ converges, by the Cauchy Criterion for series (Theorem 4.2), there exists $N \in \mathbb{N}$ such that:

Now consider the partial sums of the function series: $S_n(x) = \sum_{k=1}^n f_k(x)$. For any $m > n \ge N$ and for all $x \in E$:

By the Uniform Cauchy Criterion (Theorem 7.7), the sequence of partial sums $(S_n(x))$ converges uniformly on $E$. Therefore, $\sum_{n=1}^\infty f_n(x)$ converges uniformly on $E$. $\blacksquare$

1. Uniform Limit of Continuous Functions is Continuous

Does the limit of continuous functions have to be continuous? For pointwise convergence, the answer is NO: $f_n(x) = x^n$ on $[0, 1]$ is continuous for each $n$, but converges pointwise to $f(x) = 0$ ($x < 1$) and $f(1) = 1$, which is discontinuous at $x = 1$. Under uniform convergence, continuity is preserved!

Theorem 7.9 (Uniform Convergence Preserves Continuity): If $f_n: E \to \mathbb{R}$ is continuous on $E$ for each $n$, and $f_n \rightrightarrows f$ uniformly on $E$, then the limit function $f$ is continuous on $E$.

Rigorous "$\epsilon/3$" Proof:

Let $c \in E$ and let $\epsilon > 0$ be given. Since $f_n \rightrightarrows f$ uniformly, choose $N \in \mathbb{N}$ such that:

Since $f_N$ is continuous at $c$, choose $\delta > 0$ such that:

Now, for any $x \in E$ with $|x - c| < \delta$, apply the Triangle Inequality:

Thus $f$ is continuous at $c$. Since $c$ was arbitrary, $f$ is continuous on $E$. $\blacksquare$

2. Interchange of Limit and Integral

Theorem 7.10 (Uniform Convergence & Integration): Let $f_n \in \mathcal{R}[a, b]$ for each $n$, and suppose $f_n \rightrightarrows f$ uniformly on $[a, b]$. Then $f \in \mathcal{R}[a, b]$ and:

Proof:

Let $\epsilon > 0$. Choose $N$ such that $\forall n \ge N$ and $\forall x \in [a, b]$, $|f_n(x) - f(x)| < \frac{\epsilon}{b - a}$. Then for any $n \ge N$:

3. Interchange of Limit and Derivative

Theorem 7.11 (Uniform Convergence & Differentiation): Let $f_n: [a, b] \to \mathbb{R}$ be differentiable on $[a, b]$. Suppose:

1. There exists at least one point $x_0 \in [a, b]$ where $(f_n(x_0))$ converges.
2. The sequence of derivatives $(f'_n)$ converges uniformly on $[a, b]$ to some function $g$.

Then $(f_n)$ converges uniformly on $[a, b]$ to a differentiable function $f$, and: