Jump to content

Talk:Henri Leon Lebesgue

Page contents not supported in other languages.
From Wikipedia, the free encyclopedia

Overview of this talk page

I came across the Henri Leon Lebesgue, and found that Michael Hardy had edited and stored away on this talk page the prior entry; and on the main page substituting, in part, an explanation of his actions on the main page. I think he was right to edit the page, but think equally the discussion should take place entirely on the Talk page.

So Michael's comments are below; next is the original post; and finally some discussion.

Now all we need is for someone with knowledge of Lebesgue to happen across the page and it can all be straightened out :) --Tagishsimon 18:26, 19 Mar 2004 (UTC)

Michael Hardy's edit snipped from main page

[I have moved the rest of the content to the discussion page. On the one hand, it is clear that the author of this page knows some Lebesgue integration theory. On the other hand, there is already a page on Lebesgue integration, and most of the material belongs there rather than in an article about the person. The TeX is not correctly done. It should be like this:

rather than like this: \Gamma(\alpha)=\int_0^\infty x^{\alpha-1}\,e^{-x}\,dx, and it's worse than the latter. The author of this page doesn't appear to know how to set context. Michael Hardy 01:42 Mar 27, 2003 (UTC)

Original Article

In the early 20th century, Lebesgue introduced a new theory of integration which bridged an important gap. In order to understand this we must first point out shortcomings of the pre-existing theory of integration. In order for a function to be Riemann integrable, it must be continuous almost everywhere. Most every situation in which integrals arise, it becomes neccesary to approximate them, and so we consider a sequence of functions having an integrated value close to the approximand. Clearly, finding the limiting value of a sequence of integrals is a very popular practice! It is not hard to find a sequence of functions, each of which is Reimann integrable, whose integrated values converge, but the limiting function is not Reimann integrable. Consider f_n(x) = sum_{k=1}^{n} 1/2 (x - 1/k)^(-1/2) I(1/k <= x <= 1/k + 1/k^4) The integral on [0,1] of f_n(x) equals the sum of the first n reciprocal squares, which has a limit as n-> infinity. f_n(x) has integrable singularities at 1, 1/2, 1/3, ..., 1/n. The limit function has integrable singularities along a sequence with a limit point, and therefore is not Reiman integrable, but yet we think we know what the value of the integral should be, the infinite sum of reciprocal squares!! Thus, Lebesgue's greatest contribution to the theory of measure and integration was the developement of a new theory of integration such that the space of Lebesgue-integrable functions endowed with the integrated absolute difference metric || f - g ||_L1 = \int | f(x) - g(x) | dx is complete. That is, all limit points of Lebesgue-integrable sequences are Lebesgue integrable. This Lebesgue theory can be understood from two vantage points. First we begin by poking fun of poor Herr Reimann. Most likely he began with functions of a single variable. He knew that in order to define the integral rigorously he had to take the limit of an approximating sum. Without ever realizing that he had a choice from among the x _and_ y axes, he obfuscated the theory of integration for who knows how many decades by making the unfortunate choice of partitioning the x-axis. This is why it is hard in begining calculus to see the connection between the integral and averaging. However, the concept of averaging is also the key to understanding the Lebesgue integral. Lebesgue made the correct choice and formed an approximating sum by partitioning the y-axis. That is, let y_{n,i} = i/n for i= -n^2, -n^2+1, ..., -1, 0, 1, ..., n^2-1, n^2, and put

\int_a^b f(x) dx

= lim_{n->infinity} \int_a^b \sum_{i=1}^{n^2} y_{n,i} I(y_{n,i} <= f(x) < y_{n,i+1}) dx

= lim_{n->infinity} \sum_{i=1}^{n^2} y_{n,i} m(y_{n,i} <= f(x) < y_{n,i+1})

where m(.) is Lebesgue measure. What is Lebesgue measure? This brings us to the second vantage point. If the function is a nice function, say continuous, then the places where the function values are contained in an interval on the y-axis is a union of disjoint intervals on the x-axis. The Lebesgue measure of a union of disjoint intervals is just the ordinary notion of length. However, this is where the key ingredient to the theory arises. We need to extend the concept of length in the following way. The convergence of the above sum requires that m(.) must assign a "length" or measure to any countable union of disjoint open intervals. Our generalized notion of length should also assign a measure to the complement of any given measurable set. From this follows the requirement of measurability of any countable intersection of measurable sets. All that we have done here is to formalize intuitive notions of length by asking what is the most general type of set that can arise by taking countable unions, countable intersections, and complements of open intervals. These are the measurable sets on the real line. Once we understand measurable sets, we can understand measurable functions. A measurable function is any function whose inverse image of an open interval is a measurable set. The Lebesgue-integrable functions are all measurable functions for which the approximating sum above converges. This space of functions is a complete separable metric space under the integrated absolute difference metric, and includes such non-Reiman integrable functions such as the limit point of the above example, as well as the indicator function of the rationals and the Cantor function. Understanding measure theory is crucial to understanding probability theory.


Why is all of the dense narrative above on this page? -- Zoe

I thought someone might want to discuss how much of it to keep in the article, how much to move over to the Lebesgue integration page, and how to revise any that may remain in the article to set context and fix the TeX and otherwise make it more readable. Michael Hardy 22:01 Mar 27, 2003 (UTC)

There is an excellent (partially older) article on Lebesgue at Henri Lebesgue. This stub and that other article are combined now at the other address. -- Gauss 15:15, 19 May 2004 (UTC)[reply]