Refinement and Monotonicity of Darboux Sums

The refinement monotonicity lemma is not an isolated curiosity — it is the engine behind several downstream results students meet immediately after. It is what allows one to define the upper and lower integrals as a genuine infimum and supremum (rather than an ill-defined limit), which feeds directly into The Darboux Integral and the Riemann Integrability Criterion (a bounded function is integrable iff for every $\varepsilon > 0$ there is a partition with $U(f,P) - L(f,P) < \varepsilon$ — a statement that presupposes refining never makes the gap worse). In numerical analysis, the same monotonicity principle underlies adaptive quadrature methods, where a numerical integration routine refines a mesh locally and relies on the guarantee that finer meshes give provably tighter error bounds. In measure theory, the analogous monotonicity of upper and lower sums under refinement of a partition generalizes to monotonicity of approximating sums under refining a measurable partition, a recurring pattern in the construction of the Lebesgue integral and in probability theory's treatment of conditional expectation with respect to refining $\sigma$-algebras (filtrations). The order-theoretic idea — that a directed family of partitions under inclusion produces monotone approximating quantities — also appears in the theory of nets and Riemann–Stieltjes sums, and in computer science, in the analysis of divide-and-conquer numerical integration algorithms where correctness proofs invoke exactly this "refining never hurts" guarantee.

:::quiz[If $P \subseteq P^*$ are partitions of a bounded function $f$ on $[a,b]$, which inequality is always true?]

• $U(f,P) \le U(f,P^*)$
• $L(f,P^*) \le L(f,P)$
• $U(f,P^*) \le U(f,P)$ ✓
• $L(f,P) = L(f,P^*)$ always

:::quiz[Given two partitions $P_1, P_2$ of $[a,b]$ that are not refinements of each other, how do we prove $L(f,P_1) \le U(f,P_2)$?]

• We assume it directly from the definition of infimum and supremum
• We compare corresponding subintervals of $P_1$ and $P_2$ term by term
• We use the common refinement $P_1 \cup P_2$ and chain inequalities through it ✓
• We require $P_1$ and $P_2$ to have the same number of points

:::quiz[What role does boundedness of $f$ on $[a,b]$ play in the theorem?]

• It ensures the partition points are rational numbers
• It ensures each $M_i = \sup_{I_i} f$ and $m_i = \inf_{I_i} f$ is a finite real number so sums are well-defined ✓
• It ensures $f$ is monotonic on each subinterval
• It has no logical role; it is stated only for convention

Ask the tutor: How does this monotonicity lemma get used to prove the Riemann integrability criterion involving $U(f,P) - L(f,P) < \varepsilon$?

Picture the graph of a bounded function over $[a,b]$, with a partition drawn as vertical grid lines. Over each subinterval, the upper sum uses the tallest rectangle that still fits under (or touches) the graph's ceiling, and the lower sum uses the shortest rectangle that still reaches the graph's floor. Now insert one new grid line inside some subinterval $I_i = [x_{i-1}, x_i]$, splitting it into two pieces $[x_{i-1}, c]$ and $[c, x_i]$. The supremum of $f$ over each of the two smaller pieces can only be less than or equal to the supremum over the whole of $I_i$, because each smaller piece is a subset of $I_i$ and taking the sup over a smaller set never increases it. So the single wide rectangle of height $M_i$ is replaced by two narrower rectangles whose heights are each at most $M_i$ — the total upper area can only shrink or stay the same. Symmetrically, infima over subsets only increase, so lower sums only grow. Once this "one new point" step is justified, refining by finitely many points is just repeating the step, which is where induction enters. The final piece — that any lower sum is at most any upper sum, even across unrelated partitions — follows by comparing both to the sums for their common refinement, which sits above both partitions in the refinement order and lets us chain the inequalities together.

Example 1: Refining an asymmetric partition of a linear function. Let $f(x) = 3x + 1$ on $[0, 4]$, and let $P = \{0, 1, 4\}$ (an intentionally uneven partition — one short subinterval, one long one). Since $f$ is increasing, on any subinterval $[u,v]$ we have $\sup f = f(v) = 3v+1$ and $\inf f = f(u) = 3u+1$. Subintervals of $P$: $[0,1]$ with $\Delta x_1 = 1$, and $[1,4]$ with $\Delta x_2 = 3$.

Now refine by inserting the single point $c = 2.5$ into the long subinterval $[1,4]$, giving $P^* = \{0, 1, 2.5, 4\}$. The subinterval $[0,1]$ is unchanged. The subinterval $[1,4]$ splits into $[1, 2.5]$ (length $1.5$) and $[2.5, 4]$ (length $1.5$):

Check the theorem's chain:

Indeed $13 \le 19.75 \le 36.25 \le 43$, confirming refinement decreased the upper sum ($43 \to 36.25$) and increased the lower sum ($13 \to 19.75$), exactly as the theorem predicts — and note the decrease and increase are of different magnitudes ($6.75$ vs.\ $6.75$ here coincidentally equal since $f$ is linear with slope $3$ and the split was symmetric in length, but observe this need not happen in general, as the next example shows with a nonlinear function).

Example 2: Two unrelated partitions and the common-refinement argument, for a nonlinear function. Let $f(x) = x^2$ on $[0,3]$. Take two partitions that are not refinements of one another:

Compute $L(f,P_1)$. On $[0,2]$: $\inf f = f(0) = 0$. On $[2,3]$: $\inf f = f(2) = 4$.

Compute $U(f,P_2)$. On $[0,1]$: $\sup f = f(1) = 1$. On $[1,3]$: $\sup f = f(3) = 9$.

The theorem's corollary claims $L(f,P_1) \le U(f,P_2)$, i.e. $4 \le 19$ — true, but let's verify it is not a coincidence by explicitly building the common refinement $P^* = P_1 \cup P_2 = \{0,1,2,3\}$ and checking the full chain. Subintervals of $P^*$: $[0,1], [1,2], [2,3]$, each of length $1$.

Since $P_1 \subseteq P^*$, the theorem gives $L(f,P_1) \le L(f,P^*)$, i.e. $4 \le 5$ — true. Since $P_2 \subseteq P^*$, the theorem gives $U(f,P^*) \le U(f,P_2)$, i.e. $14 \le 19$ — true. And $L(f,P^*) \le U(f,P^*)$ gives $5 \le 14$ — true. Chaining exactly as in Step 4 of the proof:

which reproduces $4 \le 19$ but through the intermediate common refinement, exactly as the proof constructs it — not by any special relationship between $P_1$ and $P_2$ themselves.