Real Analysis 50 | Properties of the Riemann Integral for Step Functions [dark version]

For step functions, the Riemann integral behaves exactly like a well-behaved “oriented area” functional: it turns a step function into a real number while preserving the key algebraic and order structure. Once the integral is already known to be well-defined for step functions, three properties become central—scaling, additivity, and monotonicity—and together they show the integral acts like a linear, order-preserving map from step functions to real numbers. This matters because those same properties are the backbone for extending the Riemann integral to broader classes of functions.

The integral of a step function is interpreted as the oriented area between the graph and the x-axis, so the result can be positive, negative, or zero depending on whether the rectangles lie above or below the axis. Formally, the mapping takes the set of all step functions defined on an interval [a,b] (denoted S) and outputs a real number. The first major property is homogeneity: for any real scalar Λ and any step function f, the integral of the scaled function Λf equals Λ times the integral of f. Geometrically, scaling the heights of the rectangles scales the oriented area by the same factor, so the algebra matches the picture.

The second property is additivity: for two step functions f and p (written as f + p), the integral of their sum equals the sum of their integrals. This is not just a formal convenience; it reflects how rectangle areas combine when two step functions are added pointwise. The proof strategy relies on using a single partition that works for both functions simultaneously. If f is built from a partition P1 and p from a partition P2, then the combined partition P3 is taken as the union of P1 and P2. With that shared partition, both step functions are constant on each subinterval, so the Riemann sums can be aligned term-by-term.

The third property is monotonicity. If one step function lies everywhere above another—meaning f(x) ≤ p(x) for all x in the interval—then the oriented area respects that order: the integral of f is less than or equal to the integral of p. Intuitively, increasing the function height on each subinterval increases the corresponding rectangle contribution, and summing preserves the inequality.

A detailed proof is sketched for the additivity part, since the other two follow a similar pattern. The key maneuver is aligning partitions so the same subinterval endpoints are used in both Riemann sums. After that alignment, the combined sum uses (Cj + Dj) times the subinterval lengths, which corresponds exactly to the rectangle heights of the pointwise sum f + p. The result is that the Riemann sum for f + p equals the sum of the Riemann sums for f and p, yielding the desired identity for the integrals. These properties then become the template for the general Riemann integral definition in the next installment of the series.