does the result given still hold, if i replaced Darboux sums with Reimann sums?

Question

Corollary : Let $I=[a,b]$ and let $f$ be bounded on $I$. If $\{P_n : n \in \mathbb{N} \}$ is a sequence of partitions of $I$ such that $\lim_{n \to \infty} U(f;P_n)-L(f;P_n)=0$ then $f$ is integrable and $\lim_{n \to \infty} U(f;P_n) = \int_a^b f(x) \, dx = \lim_{n \to \infty} L(f;P_n)$.

In the given result, I need to understand if i replaced Darboux sums ( upper and lower sums) with Riemann sums in the limit, will the result still hold? and why isn't the converse true? even if i took $S(f;P_n)-S(f;Q_n)$ and as the difference went to zero, would it be an Riemann integrable function? I think the converse of this result should be true,saying f is integrable allows me to take any difference of sums ( either Riemann or Darboux) and will converge to zero as the mesh of $P_n$ goes to zero. Also if i took Riemann sums instead in the limit for a sequence of partitions, satisfying the condition, then f still is integrable.

Answer 1

What does work is this. Let $f$ be bounded on $I = [0,1]$. Let $P_n$ be a sequence of partitions of $I$. Let $L \in \mathbb R$.
Assume: for every choice $Q_n$ of tags for $P_n$ we have $S(P_n,Q_n,f) \to L$. Then $f$ is Riemann integrable and $\int_a^b f(x)\;dx = L$.

Explanation of notation: If $P = \{a=x_0 < x_1 < \dots < x_k = b\}$ is a partition, then a choice of tags for $P$ is a set $Q = \{t_1,t_2,\dots,t_k\}$ such that $x_{j-1} \le t_j \le x_j$ for all $j$. Then the corresponding Riemann sum is $$ S(P,Q,f) := \sum_{j=1}^k f(t_j)\;(x_j-x_{j-1}) $$

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So the point is: for this direction, we must allow all choices of tags. In the proof that it works, we choose tags with $f(t_j)$ close to $\sup\{f(x) : x_{j-1} \le x \le x_j\}$ on the one hand, and close to $\inf\{f(x) : x_{j-1} \le x \le x_j\}$ on the other hand.

For the other direction: if $f$ is Riemann integrable, then for any choice $P_n$ of partitions with norm going to zero, we can take any choice $Q_n$ of tags we like, and conclude $S(P_n,Q_n,f) \to L$. Convenient choices for $Q$ are: left endpoint; right endpoint; max point; min point. Another sneaky choice of tags (useful in certain proofs) is the one we get from the mean value theorem, $$ f(t_j) = \frac{1}{x_{j}-x_{j-1}}\int_{x_{j-1}}^{x_j} f(x)\;dx. $$

Answer 2

Your corollary is fine and is an easy consequence of the criterion for Riemann integrability :

Theorem 1: A bounded function $f:[a, b] \to\mathbb {R}$ is Riemann integrable on $[a, b] $ if and only if for every $\epsilon >0$ there is a corresponding partition $P_{\epsilon} $ of $[a, b] $ such that $U(f, P_{\epsilon}) - L(f, P_{\epsilon}) <\epsilon $.

With some more effort we can prove that:

Theorem 2: A bounded function $f:[a, b] \to\mathbb {R}$ is Riemann integrable on $[a, b] $ if and only if for every $\epsilon >0$ there is a corresponding $\delta>0 $ such that $U(f, P) - L(f, P) <\epsilon $ for every partition $P$ of $[a, b] $ with norm less than $\delta$.

In terms of sequence of partitions you now get the following result based on Theorem 2:

Theorem 3: Let $f:[a, b] \to\mathbb {R} $ be a bounded function and let $\{P_n\} $ be a sequence of partitions of $[a, b] $ such that $||P_n||$, the norm of $P_n$, tends to $0$ as $n\to\infty $. Then $f$ is Riemann integrable on $[a, b] $ if and only if $\lim_{n\to\infty} U(f, P_n) - L(f, P_n) =0$.

For Riemann sums we have the following condition which is similar to Cauchy condition:

Theorem 4: Let $f:[a, b] \to\mathbb {R} $ be a bounded function. Then $f$ is Riemann integrable on $[a, b] $ if and only if for every $\epsilon>0$ there is a corresponding partition $P_{\epsilon} $ of $[a, b] $ such that $|S(f, P), - S(f, Q)|<\epsilon$ for all partitions $P, Q$ of $[a, b] $ with $P_{\epsilon} \subseteq P, P_{\epsilon} \subseteq Q$.

There is a similar criterion for Riemann sums in terms of partition norms:

Theorem 5: Let $f:[a, b] \to\mathbb {R} $ be a bounded function. Then $f$ is Riemann integrable on $[a, b] $ if and only if for every $\epsilon>0$ there is a corresponding $\delta>0$ such that $|S(f, P), - S(f, Q)|<\epsilon$ for all partitions $P, Q$ of $[a, b] $ with norms less than $\delta$.