Monotonicity of Upper and Lower Sums

Two Sources of Monotonicity

Upper and lower sums depend both on the function being measured and on the partition used to measure it. The previous tutorial studied what happens when partition points are inserted: the established Theorem (Upper and Lower Sums Under Refinement) says that refinement cannot increase an upper sum or decrease a lower sum. Here we examine a different change. Keep the partition fixed, and compare the sums of two functions that are ordered pointwise.

The distinction matters. Refining a partition changes the intervals over which the suprema and infima are taken. Ordering two functions changes the values being bounded on those intervals. We will prove the pointwise comparison directly, then use it to obtain a quantitative estimate for functions that are uniformly close. The conclusions concern bounded functions on a closed interval and a fixed partition unless stated otherwise.

Let \(P=\{x_0,\ldots,x_n\}\) be a partition of \([a,b]\), with \(a=x_0<x_1<\cdots<x_n=b\). For a bounded function \(h:[a,b]\to\mathbb{R}\), write

In this notation,

Because the functions are bounded, every infimum and supremum in these formulas is finite. The interval lengths are positive, so an inequality between corresponding infima or suprema is preserved when multiplied by an interval length and added to the other terms.

Pointwise Order Gives Ordered Sums

Proof. Fix an interval \([x_{i-1},x_i]\) of \(P\). For every \(x\) in that interval, \(f(x)\leq g(x)\leq M_i(g)\). Thus \(M_i(g)\) is an upper bound for the values of \(f\) there, and the least-upper-bound property gives \(M_i(f)\leq M_i(g)\).

For the infima, \(m_i(f)\leq f(x)\leq g(x)\) for every \(x\) in the interval. Therefore \(m_i(f)\) is a lower bound for the values of \(g\), so \(m_i(f)\leq m_i(g)\). Multiplying each of these inequalities by the positive length \(x_i-x_{i-1}\) and summing over \(i=1,\ldots,n\) gives

and

These are exactly the asserted inequalities for the lower and upper sums. \(\square\)

This theorem does not require the functions to be continuous or to attain their extrema. It compares infima with infima and suprema with suprema on each same interval. In particular, the conclusion does not say that a lower sum for one function must be smaller than an upper sum for another for the reason just proved; comparisons involving different partitions use the earlier Theorem (Comparison Across Partitions).

Uniform Closeness Controls Both Sums

Pointwise order also provides a useful way to compare functions that are close but not ordered in one direction. If their values differ by at most a fixed amount, each function is bounded above and below by a vertical shift of the other. This yields a bound for each sum that depends only on the size of the shift and the total interval length.

Proof. The hypothesis gives \(f(x)\leq g(x)+\delta\) and \(g(x)\leq f(x)+\delta\) for all \(x\). On any partition interval, adding a constant \(\delta\) to a function adds \(\delta\) to its infimum and to its supremum. This follows because shifting every value by \(\delta\) shifts every upper bound and every lower bound by the same amount. Consequently, the pointwise-order theorem gives

Here the added \(\delta\) contributes \(\delta\) times the sum of the partition interval lengths, and

Together, the two upper-sum inequalities imply \(|U(f,P)-U(g,P)|\leq\delta(b-a)\). Applying the same pointwise-order theorem to lower sums gives

which implies \(|L(f,P)-L(g,P)|\leq\delta(b-a)\). This proves both bounds. \(\square\)

The same estimates also control the Darboux gaps. Subtracting lower sums from upper sums and using the triangle inequality gives

The factor of \(2\) arises because both the upper sum and the lower sum may change by as much as \(\delta(b-a)\), in opposite directions. This is a bound, not a claim that the gap always changes by that amount.

Using the Two Monotonicity Principles Together

The two kinds of monotonicity can be applied in sequence. If \(f\leq g\), then for a fixed partition \(P\), the pointwise-order theorem compares their sums. If \(Q\) is a refinement of \(P\), the established Theorem (Upper and Lower Sums Under Refinement) compares sums for the same function across those partitions. Each comparison has its own hypothesis, so it is important to keep the function and partition being changed clear.

For example, suppose \(f\leq g\) and \(Q\) refines \(P\). The results give

and hence \(L(f,P)\leq L(g,Q)\). For upper sums they give

so \(U(f,Q)\leq U(g,P)\). The direction of the refinement comparison differs for lower and upper sums; reversing one of those directions is a common source of mistakes.

A second useful application is to enclose sums of a function between those of simpler bounding functions. If \(h\leq f\leq k\), the pointwise-order theorem gives \(L(h,P)\leq L(f,P)\leq L(k,P)\) and \(U(h,P)\leq U(f,P)\leq U(k,P)\). Such bounds can be combined with the Darboux Criterion when choosing a partition: estimates on the functions provide estimates on both sums without requiring the extrema of \(f\) itself to be computed exactly.

Remember that all suprema and infima are taken over the whole closed partition interval. The values at endpoints are included, whether or not a supremum or infimum is attained elsewhere. Also, the uniform perturbation theorem requires one common bound \(\delta\) valid at every point. A pointwise statement that the difference is small “near most points” does not by itself supply the hypothesis.