
Annotated version of Lebesgue’s 1901 paper on the integral - fermatslibrary
http://fermatslibrary.com/s/on-a-generalization-of-the-definite-integral
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adenadel
When Fermat's Library started I really eagerly signed up. There are some
usability issues with the interface, but my main disappointment is with the
annotations. I would be really interested not in crowdsourced annotations, but
in annotations and context given by an expert in the field (or perhaps one of
the authors if possible) trying to present the paper as a story to a lay/semi-
technical audience.

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dorfsmay
> Lebesgue’s 1901 paper that changed the integral . . . forever

I though that Lebesgue "just" proved that the method scientists had been using
for a while already was mathematically sound? Has it really changed the
calculus landscape?

~~~
verylongname
Lebesgue integrals differ in several key respects from Riemann integrals.
Intuitively, Riemann integrals involve partitioning the domain of a function
into disjoint intervals

(a_1,a_2), (a_2,a_3), ... (a_{n-1},a_n)

and approximately the area under the graph of the function via sums of the
form

\sum_i f(x_i) (a_{i+1} - a_i)

where x_i is a point in the interval (a_i,a_{i+1}).

Lebesgue integrals turn this procedure on its head by partitioning the range
of f into disjoint intervals

(a_0,a_1), (a_1, a_2), ... (a_{n-1},a_n)

and approximating the area under the graph of the function via

\sum_i m({x : a_i < f(x}) < a_{i+1}) a_i

where m(E) refers to the "measure" of the set E. That is, for each i, we
multiply the "size of the set on which f is mapped to a value near a_i" by a_i
and then sum over i.

The principle advantage of the Lebesgue scheme is that f can be very badly
behaved and the quantities involved are still well-defined and make sense,
whereas the Riemann integral only leads to reasonable approximations if f is
somewhat well-behaved (more-or-less continuous). Otherwise, the value of
f(x_i) (x_{i+1}-x_i) is not a reasonable approximation of the area under the
graph of f "over the interval (x_{i+1}-x_i)".

There are even more general notions of integral. To my knowledge, most are
based on observing that an integral is a linear functional on some space which
should satisfy certain properties.

~~~
woopwoop
This is a nice summary, but I've always found the characterization of Riemann
integrals as "partitioning the domain" and Lebesgue integrals as "partitioning
the range" unsatisfying. This is mainly an artifice of the common
constructions, but one can give definitions of the Riemann and Lebesgue
integrals where the only difference is that, in several places, one must
replace the word "finite" with the word "countable". Here is one such:

The definition of the length of an interval should be obvious. Let I be an
interval, and let E be a subset of I. Define its Jordan outer measure to be
the inf of the sums of the lengths of finite collections of intervals covering
E. Define its Jordan inner measure to be the length of I minus the Jordan
outer measure of I \setminus E. E is called Jordan measurable if its outer and
inner Jordan measures are equal. A function s is Jordan simple if it is a
linear combination of characteristic functions of Jordan measurable sets.
Define the integral of Jordan simple functions in the obvious way. A bounded
function on I is Riemann integrable if and only if it is the uniform limit of
Jordan simple functions, and its Riemann integral is the limit of the
integrals of the approximating simple functions.

If, in the previous paragraph, one replaces the word "finite" with the word
"countable", and the names "Jordan" and "Riemann" with "Lebesgue", one
recovers the Lebesgue integral.

~~~
verylongname
You are just defining the class of Riemann integrable and Lebesgue integrable
functions using Jordan measurability. There is nothing wrong in this, but
there is nothing different in it either. It is equivalent to the standard
limit of simple functions definitions.

~~~
woopwoop
I agree, I was just pointing out that it is possible to define both the
Riemann and Lebesgue integrals by "partitioning the range". The real
difference lies in the choice to allow countable rather than finite covers by
intervals.

