> 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?
For applications in physics and other sciences the Lebesgue integral is not critical. Even if you care about making sure your calculations are mathematically sound then the theory of the Riemann integral (much older than Lebesgue's) is sufficient.
However in mathematics, it did change how people think about integration. "Integral" now usually means "Lebesgue integral" outside of specialized applications. One of its advantages is that a lot of theorems involving integration become simpler to state, since with the Riemann integral you need to add more conditions to make sure it is defined.
To admit delta functions (e.g., impulse responses) you need at least a Riemann-Stieltjes integral. And that is (as far as I know) something of a patch.
Conventional Hilbert spaces (e.g., for QM) are defined with respect to the Lebesgue integral. If you restrict to just Riemann-integrable functions, the Hilbert space isn't complete (IIRC) because not enough functions qualify.
Besides all this, you can argue that the Lebesgue integral is necessary to physics because it is the best known foundation for probability theory, which (again, ultimately, due to the observable manifestations of QM) is central to our understanding of the physical world. E.g., ergodic theory, Brownian motion, magnetism, phase transitions, and on and on.
IIRC, a Hilbert space cannot contain the Dirac delta function regardless of the integral used, since the inner product of the delta function with itself is ill-defined. So what people use in QM is technically "rigged Hilbert spaces" (or Gelfand triples) which a few mathematicians go to great pains to formalise and put on a rigorous foundations. But about which exactly zero shits are given by all the physics departments of the world; they simply handwave about with some box regularizations, swap integrals and limits at their hearts desire, and everything turns out fine in the end.
Yes, I wasn't meaning to imply delta "functions" are elements of Hilbert spaces. My first paragraph was just about delta functions as tools for representing impulse responses for mechanical or electrical systems, not QM.
Well, Lebesgue integration cannot handle delta functions either since they are not even functions but distributions.
(Incidentally, there is such a thing as Lebesgue-Stieltjes integral, the kind of issues Riemann-Stieltjes solve are orthogonal to the issues that motivate Lebesgue integration.)
As for the foundations of probability theory I am skeptical. I agree the theorems are nicer if you base probability theory on the Lebesgue integral, but I very much doubt if there is a realistic physical question that could not be answered by probability theory based on the Riemann integral.
And for what it is worth, a lot of modern physics (i.e., all of quantum field theory and modern statistical physics) is based on a type of integral (the path integral) that does not yet have rigorous mathematical foundations.
Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether, say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.
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.
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.
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.
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.
The main advantage of the Lebesgue integral isn't its generality, but the convergence theorems which give you L^p spaces. The gauge integral, despite being more general, doesn't have these properties and nobody uses it.
But the only obstruction to developing convergence theorems for Riemann integrals is that the limit of a sequence of Riemann integrable functions need not be Riemann integrable. This, of course, follows from the standard convergence theorems for Lebesgue integrals and the fact that the two notions coincide where both are defined. So it really does come down to the class of functions which are integrable.
Lebesgue's theory of integration is built on measure theory, which was developed by Lebesgue and others. Measure theory has turned out to be a very natural framework for probability, stochastic processes, ergodic theory, etc. (See e.g. https://en.wikipedia.org/wiki/Measure_%28mathematics%29.)
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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[ 4.4 ms ] story [ 12.5 ms ] threadI 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?
However in mathematics, it did change how people think about integration. "Integral" now usually means "Lebesgue integral" outside of specialized applications. One of its advantages is that a lot of theorems involving integration become simpler to state, since with the Riemann integral you need to add more conditions to make sure it is defined.
Conventional Hilbert spaces (e.g., for QM) are defined with respect to the Lebesgue integral. If you restrict to just Riemann-integrable functions, the Hilbert space isn't complete (IIRC) because not enough functions qualify.
Besides all this, you can argue that the Lebesgue integral is necessary to physics because it is the best known foundation for probability theory, which (again, ultimately, due to the observable manifestations of QM) is central to our understanding of the physical world. E.g., ergodic theory, Brownian motion, magnetism, phase transitions, and on and on.
(Incidentally, there is such a thing as Lebesgue-Stieltjes integral, the kind of issues Riemann-Stieltjes solve are orthogonal to the issues that motivate Lebesgue integration.)
As for the foundations of probability theory I am skeptical. I agree the theorems are nicer if you base probability theory on the Lebesgue integral, but I very much doubt if there is a realistic physical question that could not be answered by probability theory based on the Riemann integral.
And for what it is worth, a lot of modern physics (i.e., all of quantum field theory and modern statistical physics) is based on a type of integral (the path integral) that does not yet have rigorous mathematical foundations.
-- Richard Hamming
(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.
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.