nLab
Henstock integral

The Henstock integral

The Henstock integral

Idea

The Henstock integral (also attributed to Kurzweil, Denjoy, Luzin, and Perron, and sometimes called, neutrally but perhaps ambiguously, the gauge integral) is a way to define the integral of a (partial) function f:f:\mathbb{R}\to \mathbb{R} which applies to more functions than either the Riemann integral or the Lebesgue integral and is in some ways better behaved as well.

For instance, the (even) function

tsin(1/t 3)t,t{0} t\mapsto \frac{\sin(1/t^3)}{t},\quad t \in \mathbb{R}\setminus\{0\}

is not Lebesgue integrable on any interval containing 0, but it has (DR: according to WolframAlpha) Henstock integral

0 xsin(1/t 3)t=13(π2Si(1/x 3)) \int_{0}^x \frac{\sin (1/t^3)}{t} = \frac{1}{3}\left( \pi - 2 Si(1/x^3)\right)

where Si(x)Si(x) is the sine integral? 0 xsin(t)tdt\int_0^x \frac{\sin(t)}{t}dt (note that SiSi extends to an entire function on \mathbb{C}).

However, the Lebesgue integral is more commonly used by working mathematicians because it fits more naturally into the general theory of measure, while the Riemann/Darboux integral is more commonly used in introductory calculus courses because its definition is simpler.

Definition

Let [a,b][a,b] be a closed interval in \mathbb{R} and let f:[a,b]f:[a,b]\to \mathbb{R}. A tagged partition PP of [a,b][a,b] is a finite sequence of points a=u 0<u 1<<u n+1=ba = u_0 \lt u_1 \lt \dots \lt u_{n+1} = b together with points t i[u i,u i+1]t_i \in [u_i, u_{i+1}] for all ii. The Riemann sum of ff over such a tagged partition is

Pf= i=0 nf(t i)(u i+1u i). \sum_P f = \sum_{i=0}^{n} f(t_i) \cdot (u_{i+1} - u_{i}).

Define a gauge on [a,b][a,b] to be any function δ:[a,b](0,)\delta: [a,b] \to (0,\infty). We say that a tagged partition is δ\delta-fine if [u i,u i+1](t iδ(t i),t i+δ(t i))[u_i, u_{i+1}] \subset (t_i - \delta(t_i), t_i + \delta(t_i)).

Finally, we say that II is the integral of ff on [a,b][a,b], written I= a bf(x)dxI = \int_{a}^b f(x) d x, if for any ϵ>0\epsilon\gt 0 there exists a gauge δ\delta such that

| PfI|<ϵ {| \sum_P f - I |} \lt \epsilon

for any δ\delta-fine partition PP. If such an II exists, we say that ff is (Henstock) integrable on [a,b][a,b].

If we require a gauge to be a constant function, then we recover the definition of the Riemann integral.

The Henstock integral may be seen as a non-uniform generalization of the Riemann integral. Whereas specifying a constant δ\delta is tantamount to picking an entourage on [a,b][a,b], specifying a gauge δ\delta is tantamount to assigning a neighbourhood to each point in [a,b][a,b]. (Indeed, with either definition of integral, it would be equivalent to replace δ\delta in the definition with an entourage or an assignment of neighbourhoods.) Similarly, the definition of uniformly continuous function becomes that of continuous function if you change δ\delta from a constant to a gauge.

In constructive mathematics, we must allow a gauge to take lower semicontinuous values. (This is not necessary with the Riemann integral.) Otherwise, there may not be enough gauges, since these are rarely continuous. (The definition could also be made constructive by explicitly referring to an assignment of neighbourhoods to points or by replacing δ\delta with an entire relation. Again compare the change from the definition of uniformly continuous to pointwise-continuous function.)

Properties

The fundamental theorem of calculus

The Henstock integral satisfies a very nice form of the fundamental theorem of calculus:

Theorem

If ff is differentiable on [a,b][a,b], then ff' is Henstock integrable on [a,b][a,b], and a bf(x)dx=f(b)f(a)\int_a^b f'(x) d x = f(b) - f(a).

Theorem

If ff is Henstock integrable on [a,b][a,b], then F(x)= a xf(t)dtF(x) = \int_a^x f(t) d t is differentiable almost everywhere on [a,b][a,b] and F=fF' = f.

Hake’s theorem

Theorem

For any ff we have

a bf(x)dx=lim cb a cf(x)dx \int_a^b f(x) d x = \lim_{c\to b^-} \int_a^c f(x) d x

in the strong sense that if either side exists, then so does the other, and they are equal.

In particular, what is often taken as a definition of the improper? Riemann integral (of a potentially unbounded function on a finite interval) is actually a theorem for Henstock integrals. (However, we still need improper Henstock integrals to allow a=a = -\infty or b=b = \infty.)

Recovery of Riemann and Lebesgue integrals

I need to check some of the claims below, but I'm out of time right now. They are definitely correct for the proper integrals. —Toby

Recall that f:[a,b]f\colon [a, b] \to \mathbb{R} is Riemann integrable iff ff is continuous almost everywhere and bounded; in this case, ff is also Henstock integrable, and the Riemann integral of ff equals its Henstock integral.

More generally, ff is improperly Riemann integrable iff ff is Henstock integrable and ff is locally Riemann integrable at all but finitely many points in [a,b][a, b]; then the improper Riemann integral of ff equals its Henstock integral.

Still more generally, f:f\colon \mathbb{R} \to \mathbb{R} is improperly Riemann integrable iff ff is improperly Henstock integrable (meaning merely that

lim a,b a bf(x)dx \lim_{a \to -\infty, b \to \infty} \int_a^b f(x) \,\mathrm{d}x

exists using Henstock integrals) and locally Riemann integrable except at a set of isolated point?s; then the improper Riemann integral of ff equals its improper Henstock integral.

Finally (and with incomparable generality), f:f\colon \mathbb{R} \to \mathbb{R} is Lebesgue integrable iff |f|{|f|} is improperly Henstock integrable; then the Lebesgue integral of ff equals its improper Henstock integral (which is proper if the support of ff is bounded, of course).

Last revised on April 16, 2016 at 10:01:16. See the history of this page for a list of all contributions to it.