Real analysis: if two Riemann integrable functions have equivalent lower sums for any partition, their integrals are the same on any subinterval.

Question

I have to prove that if $f,g : [a,b]\longrightarrow \mathbb{R}$ are Riemann integrable functions such that $\underline{S}(f,P)=\underline{S}(g,P)$ for any partition $P$ of $[a,b]$, then $\int\limits_{r}^{s}f=\int\limits_{r}^{s}g$ for any $r,s \in [a,b], r<s$. Here's what I have tried: Trivially, $\int\limits_{a}^{b}f=\int\limits_{a}^{b}g$, and also I proved that both $f$ and $g$ are integrable on $[r,s]$ and $$ \int\limits_{a}^{b}f=\int\limits_{a}^{r}f+\int\limits_{r}^{s}f+\int\limits_{s}^{b}f=\int\limits_{a}^{r}g+\int\limits_{r}^{s}g+\int\limits_{s}^{b}g=\int\limits_{a}^{b}g$$ Also, there exists a sequence of partitions $P_n$ such that both $r$ and $s$ are the limits of intervals in $P_n$, and the differences of upper sums minus lower sums of both $f$ and $g$ of the corresponding partitions of the intervals $[a,r]$,$[r,s]$ and $[s,b]$ (partitions which are subsets of $P_n$) are all less than any $\varepsilon>0$. I now want to express the integral $\int\limits_{r}^{s}g$ as the limit of lower sums of $f$ using this sequence of partitions, but have been stuck for a while trying to figure out how. Any ideas on how to develop my idea or solve the problem in any other way would be appreciated.

Answer 1

An equivalent statement is that $\underline S(f-g,P) = 0$ for every $P$.

So what does it say about a function $h$ that $\underline S(h,P) = 0$ for every $P$?

Now $\underline S(h,[a,b]) = (b-a)\inf_{[a,b]} h = 0$, so $\inf_{[a,b]} h = 0$. Suppose there is an $c \in (a,b)$ with $\inf_{[a,c]} h \ne 0$. That would mean $\inf_{[c,b]} h = 0$, since $\inf_{[a,b]} h = \min\{\inf_{[a,c]} h, \inf_{[c,b]} h\}$. But then $$\underline S(h, \{[a,c],[c,b]\}) = (c-a)\inf_{[a,c]} h + (b-c) 0 \ne 0$$

which cannot be. So for every $c \in (a,b), \inf_{[a,c]} h = 0$. And similarly $\inf_{[c,b]} h = 0$. The same calculation again shows that for any $[c,d] \subset (a,b), \inf_{[c,d]} h = 0$.

Therefore it is not just any partition $P$ of $[a,b]$ for which $\underline S(h,P) = 0$, but also any partition of any sub interval of $[a,b]$, including $[r,s]$.