The Riemann-Darboux Integral

We start by recalling the definition of the familiar Riemann–Darboux integral of the calculus, which for brevity we will call the Riemann integral. Our later development of the Lebesgue integral will closely parallel this treatment of the Riemann integral. We consider a fixed bounded interval [a, b] and consider only real functions f which are bounded on [a,b].

A partition P of [a, b] is a set P = {x0, x1, x2,…, xn} of points of [a, b] with

=x0<x1<x2<…<xn=b.

Let f be a given function on [a, b] with

≤f(x)≤M

for all x ∈ [a, b]. For each i = 1, 2,…, n let

i=inf⁡{f(x):xi−1<x<xi},Mi=sup⁡{f(x):xi−1<x<xi}.

In the usual calculus text treatment the infs and sups are taken over the closed intervals [xi−1, xi]. In our treatment of the Lebesgue integral we will partition [a, b] into disjoint sets, so we use the disjoint sets (xi−1, xi) here. We are effectively ignoring a finite set of function values, f(x0), f(x1), f(x2),…, f(xn), and in so doing we anticipate the important result of Lebesgue that function values on a set of measure zero (here the set of partition points) are not relevant for either the Riemann or Lebesgue integral.

The lower sum L(f, P) and the upper sum U(f, P) for the function f and the partition P are defined as follows:

(f,P)=∑i=1nmi(xi−xi−1),U(f,P)=∑i=1nMi(xi−xi−1).

Clearly m ≤ mi ≤ Mi ≤ M for each i, so

(b−a)≤L(f,P)≤U(f,P)≤M(b−a).

For a positive function f on [a, b] the lower sum represents the sum of the areas of disjoint rectangular regions which lie within the region

={(x,y):a≤x≤b,0≤y≤f(x)}.

Similarly, the upper sum U(f, P) is the area of a finite number of disjoint rectangular regions which cover the region S except for a finite number of line segments on the lines x = xi. The function f is said to be integrable whenever

PL(f,P)=inf⁡PU(f,P).

The integral of f over [a, b] is the common value in (1) and is denoted abf. The area of S is defined to be this integral whenever f is integrable and non-negative.

The integral is also sometimes written abf(x)dx, particularly when a change of variable is involved. The “x” in this expression is a dummy variable and can be replaced by any variable except f or d. For example,

abf(x)dx=∫abf(t)dt=∫abf(a)da=∫abf(c)dc.

The last two versions are logically correct, but immoral, since they flout the traditional roles of a, b, c as constants, and the third integral discourteously uses the same letter for the limit and the dummy variable.

Now we make a few computations to derive the basic properties of the integral and show that the integral exists at least when f is continuous.

A partition Q is a refinement of the partition P provided each point of P is a point of Q. We will indicate this by writing P Q without reference to the numbering of the points in P or Q. Clearly Q is a refinement of P provided each of the subintervals of [a, b] determined by Q is contained in one of the subintervals determined by P.