504 17. REAL AND COMPLEX ANALYSISjustified his more general integral in the following words, from the preface to his
1904 monograph.
[I]f we wished to limit ourselves always to these good [that is,
smooth] functions, we would have to give up on the solution of a
number of easily stated problems that have been open for a long
time. It was the solution of these problems, rather than a love of
complications, that caused me to introduce in this book a definition
of the integral that is more general than that of Riemann and
contains the latter as a special case.Despite its complexity—to develop it with proofs takes four or five times as
long as developing the Riemann integral—the Lebesgue integral was included in
textbooks as early as 1907: for example, Theory of Functions of a Real Variable,
by E. W. Hobson (1856-1933). Its chief attraction was the greater generality of
the conditions under which it allowed termwise integration. Following the typical
pattern of development in real analysis, the Lebesgue integral soon generated new
questions. The Hungarian mathematician Frigyes Riesz (1880-1956) introduced
the classes now known as Lp-spaces, the spaces of measurable functions^13 / for
which |/|p is Lebesgue integrable, ñ > 0. (The space Loo consists of functions that
are bounded on a set whose complement has measure zero.) How the Fourier series
and integrals of functions in these spaces behave became a matter of great interest,
and a number of questions were raised. For example, in his 1915 dissertation at the
University of Moscow, Nikolai Nikolaevich Luzin (1883-1950) posed the conjecture
that the Fourier series of a square-integrable function converges except on a set of
measure zero. Fifty years elapsed before this conjecture was proved by the Swedish
mathematician Lennart Carleson (b. 1928).
2.5. The abstract and the concrete. The increasing generality allowed by the
notation y = f(x) threatened to carry mathematics off into stratospheric heights
of abstraction. Although Ampere had tried to show that a continuous function is
differentiable at most points, the attempt was doomed to failure. Bolzano con-
structed a "sawtooth" function in 1817 that was continuous, yet had no derivative
at any point. Weierstrass later used an absolutely convergent trigonometric series to
achieve the same result,^14 and a young Italian mathematician Salvatore Pincherle
(1853-1936), who took Weierstrass' course in 1877-1888, wrote a treatise in 1880 in
which he gave a very simple example of such a function (Bottazzini, 1986, p. 286):
Volterra's example of a continuous function whose derivative was not integrable,
together with the examples of continuous functions having no derivative at any
point naturally cast some doubt on the applicability of the abstract concept of
continuity and even the abstract concept of a function. Besides the construction of
more general integrals and the consequent ability to "measure" more complicated
(^1) See below for the definition.
(^14) This example was communicated by his student Paul du Bois-Reymond (1831-1889) in 1875.
The following year du Bois-Reymond constructed a continuous periodic function whose Fourier
series failed to converge at a set of points that came arbitrarily close to every point.
oo