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Definition 7.1 in Apostol's book gives the following definition for the Riemann integral. Let $P=\{x_0,x_1,\ldots,x_n\}$ be a partition of $[a,b]$ and $t_k$ any point in $[x_{k-1},x_k]$. Denote $S(P,f)=\sum_{k=1}^n f(t_k)(x_k-x_{k-1})$.

Definition 1: We say that $f$ is Riemann integrable on $[a,b]$ if there exists a real number $A$ such that, for all $\epsilon>0$, there exists a partition $P_\epsilon$ of $[a,b]$ such that, for each finer partition $P$ and for each arbitrary choice $t_k\in [x_{k-1},x_k]$, it holds $|S(P,f)-A|<\epsilon$.

One of the intuitions I have to better understand the Riemann integral could be summarized in the following definition:

Definition 2: We say that $f$ is Riemann integrable on $[a,b]$ if there exists a real number $A$ and a sequence of partitions $\{P_n\}$ with mesh tending to $0$, such that, for each arbitrary choice $t_k\in [x_{k-1},x_k]$ in $P_n$, it holds $\lim_{n\rightarrow\infty} S(P_n,f)=A$.

I have never seen a proof of the equivalence of both definitions. Maybe they are not equivalent.

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The definitions are equivalent.

We have the Riemann criterion whereby a function $f$ is integrable over $[a,b]$ according to Definition 1 if and only if for every $\epsilon > 0$ there exists a partition $P$ such that upper and lower sums satisfy $U(P,f) - L(P,f) < \epsilon$.

Suppose $f$ satisfies Definition 2. Given $\epsilon > 0$ there exists a positive integer $N$ such that for all $n \geqslant N$ and any choice of tags $\{t_j\}$ we have $|S(P_n,f, \{t_j\}) - A| < \epsilon/4$. In particular, we have

$$\tag{*} A - \epsilon/4 < S(P_N,f,\{t_j\}) < A + \epsilon/4.$$

Consider any subinterval $I_j = [x_{j-1},x_j]$ of $P_N$. Let $m_j = \inf_{x \in I_j} f(x)$ and $M_j = \sup_{x \in I_j}f(x)$.

There exist points $\alpha_j, \beta_j \in I_j$ such that

$$m_j \leqslant f(\alpha_j) < m_j + \frac{\epsilon}{4(b-a)}, \\ M_j - \frac{\epsilon}{4(b-a)} < f(\beta_j) \leqslant M_j. $$

Multiplying by $(x_j - x_{j-1})$, summing over $j$ and using (*) we get

$$A - \epsilon/4<S(P_N,f, \{\alpha_j\})< L(P_N,f) + \epsilon/4, \\ U(P_N,f) - \epsilon/4 < S(P_N,f, \{\beta_j\})< A + \epsilon/4$$

This implies $A- \epsilon/2 < L(P_N,f)$ and $U(P_N,f) < A + \epsilon/2$. and, hence,

$$U(P_N,f) - L(P_N,f) < \epsilon.$$

Since we are able to find a partition where the Riemann criterion is satisfied, a function that is integrable under Definition 2 is also integrable under Definition 1.

The converse is easy to prove.

Addendum

There are two possible interpretations of $\lim_{n \to \infty} S(P_n,f) = A$ in Definition 2.

(2a) For every $\epsilon > 0$ there exists a positive integer $N(\epsilon)$ such that if $n \geqslant N(\epsilon)$, then $|S(P_nf, \{t_j\})-A| < \epsilon$ holds for every choice of tags $\{t_j\}$.

(2b) For every $\epsilon > 0$ and each choice of tags $\{t_j\}$, there exists a positive integer $N(\epsilon, \{t_j\})$ such that if $n \geqslant N(\epsilon, \{t_j\})$, then $|S(P_nf, \{t_j\})-A| < \epsilon$ holds.

To be precise, Definition 1 is equivalent to Definition 2a. It was shown above that (2a) implies (1). The converse follows because if a function is Riemann integrable under Definition 1, there is an equivalence to the statement that for any $\epsilon > 0$ there exists $\delta > 0$ such that $\|P\| < \delta $ implies that $|S(P,f, \{t_j\}) - A| < \epsilon$ for any choice of tags.

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  • $\begingroup$ Why is the number $N$ the same for all tags $\{t_j\}$? A priori, the rapidness of convergence depends on the tag. $\endgroup$ – user39756 Feb 5 '18 at 19:01
  • $\begingroup$ @user39756: That's the way I'm reading the statement and that is what I'm proving. You won't find this definition 2 in any book that I've seen so I'm not sure how convoluted it should be. Normally the alternative definition to integrability in terms of partition refinement is that, for every $\epsilon> 0$ there is a $\delta >0$ such that for any partition $P$ with mesh $\|P\| < \delta$ we have $|S(P,f,T) - I| < \epsilon$ for any choice of tags. Even here books will chose one definition over another and never prove the equivalence. $\endgroup$ – RRL Feb 5 '18 at 19:12
  • $\begingroup$ Where did you get definition 2? $\endgroup$ – RRL Feb 5 '18 at 19:13
  • $\begingroup$ Under Definition 1 which I know is standard we have $U(P,f) - L(P,f) < \epsilon$ for any further refinement of the partition. All Riemann sums regardless of tags are squeezed in between upper and lower sums so all converge at the same rate. Riemann integration is "convergence of a net". I would doubt the equivalence of another definition without this property. $\endgroup$ – RRL Feb 5 '18 at 19:17
  • $\begingroup$ I think the argument would work with minor changes. We have $A$ and the partitions $\{P_n\}$. For each $\epsilon>0$ and each sequence of tags $\{t_j\}$ for each $P_n$, $n\geq1$, there exists a number $N=N(\epsilon,\{t_j\})$ such that, for all $n\geq N$, $A-\epsilon/4<S(P_n,f,\{t_j\})<A+\epsilon/4$. Take $\alpha_j$ and $\beta_j$ as you did. For the sequence of tags $\{\alpha_j\}$, there is a number $N_1(\epsilon,\{\alpha_j\})$; for the sequence of tags $\{\beta_j\}$, there is a number $N_2(\epsilon,\{\alpha_j\})$. Take $N_0=\max\{N_1,N_2\}$. Then $U(P_{N_0},f)-L(P_{N_0},f)<\epsilon$. $\endgroup$ – user39756 Feb 5 '18 at 19:28

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