请求解析黎曼积分的两个等价定义及Apostol书中的定义7.1

Alright, let's break down the two equivalent definitions of the Riemann integral thoroughly, starting with the one referenced from Apostol's Mathematical Analysis (Definition 7.1) and pairing it with the classic upper/lower integral formulation. We'll also walk through their equivalence proof step by step.

1.1 Apostol's Definition (Uniform Convergence of Riemann Sums) #

First, let's formalize the foundational concepts and complete the definition you mentioned:

Let $f$ be a bounded function on the closed interval $[a,b]$. A partition of $[a,b]$ is a finite set $P = {x_0, x_1, ..., x_n}$ where $a = x_0 < x_1 < ... < x_n = b$. For each subinterval $[x_{k-1}, x_k]$, choose any sample point $t_k \in [x_{k-1}, x_k]$. The Riemann sum of $f$ over $P$ is defined as:
$$S(P,f) = \sum_{k=1}^n f(t_k) \Delta x_k$$
where $\Delta x_k = x_k - x_{k-1}$ denotes the width of the $k$-th subinterval.

Complete Definition 1: #

A function $f$ is Riemann integrable on $[a,b]$ if there exists a real number $A$ such that:
For every $\epsilon > 0$, there exists a partition $P_\epsilon$ of $[a,b]$ where, for all partitions $P$ finer than $P_\epsilon$ (i.e., $P_\epsilon \subseteq P$) and any choice of sample points $t_k \in [x_{k-1}, x_k]$, the inequality
$$|S(P,f) - A| < \epsilon$$
holds. The number $A$ is called the Riemann integral of $f$ over $[a,b]$, written as $\int_a^b f(x) dx = A$.

1.2 Classic Upper/Lower Integral Definition #

This definition relies on bounds of the function over subintervals, rather than arbitrary sample points:

For any partition $P$ of $[a,b]$, define:

• $M_k = \sup{f(x) \mid x \in [x_{k-1}, x_k]}$ (the supremum of $f$ on the $k$-th subinterval)
• $m_k = \inf{f(x) \mid x \in [x_{k-1}, x_k]}$ (the infimum of $f$ on the $k$-th subinterval)

Then we define:

• Upper sum: $U(P,f) = \sum_{k=1}^n M_k \Delta x_k$
• Lower sum: $L(P,f) = \sum_{k=1}^n m_k \Delta x_k$

Using these sums, we define the upper integral and lower integral:

• Upper integral: $\overline{\int}_a^b f(x) dx = \inf{U(P,f) \mid P \text{ is a partition of } [a,b]}$
• Lower integral: $\underline{\int}_a^b f(x) dx = \sup{L(P,f) \mid P \text{ is a partition of } [a,b]}$

Definition 2: #

A function $f$ is Riemann integrable on $[a,b]$ if its upper integral equals its lower integral:
$$\overline{\int}_a^b f(x) dx = \underline{\int}_a^b f(x) dx = A$$
Here, $A$ is the Riemann integral of $f$ over $[a,b]$.

We need to show that Definition 1 holds if and only if Definition 2 holds.

2.1 Definition 1 ⇒ Definition 2 #

Suppose $f$ is integrable by Definition 1, with integral $A$. Let $\epsilon > 0$ be arbitrary. By Definition 1, there exists a partition $P_\epsilon$ such that for all finer partitions $P$, every Riemann sum $S(P,f)$ satisfies $A - \epsilon < S(P,f) < A + \epsilon$.

For this partition $P$:

• The lower sum $L(P,f)$ is the infimum of all possible Riemann sums over $P$, so $L(P,f) \geq A - \epsilon$
• The upper sum $U(P,f)$ is the supremum of all possible Riemann sums over $P$, so $U(P,f) \leq A + \epsilon$

By definition of upper/lower integrals:
$$A - \epsilon \leq \underline{\int}_a^b f \leq \overline{\int}_a^b f \leq A + \epsilon$$
Since $\epsilon$ can be made arbitrarily small, we must have $\underline{\int}_a^b f = \overline{\int}_a^b f = A$, so $f$ satisfies Definition 2.

2.2 Definition 2 ⇒ Definition 1 #

Suppose $f$ is integrable by Definition 2, with $\overline{\int}_a^b f = \underline{\int}_a^b f = A$. Let $\epsilon > 0$ be arbitrary.

By definition of upper/lower integrals:

• There exists a partition $P_1$ such that $U(P_1,f) < A + \frac{\epsilon}{2}$
• There exists a partition $P_2$ such that $L(P_2,f) > A - \frac{\epsilon}{2}$

Let $P_\epsilon = P_1 \cup P_2$ (the common refinement of $P_1$ and $P_2$). A key property of upper/lower sums is that refining a partition does not increase the upper sum or decrease the lower sum. So for any partition $P$ finer than $P_\epsilon$:
$$A - \frac{\epsilon}{2} < L(P_2,f) \leq L(P,f) \leq S(P,f) \leq U(P,f) \leq U(P_1,f) < A + \frac{\epsilon}{2}$$
This implies $|S(P,f) - A| < \frac{\epsilon}{2} < \epsilon$, which satisfies the conditions of Definition 1.

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