Xrfgtjtk

How would you describe the material of measure theory as a chain of thoughts which each gave birth to another? How could you describe the thought process that went through the minds of mathematical inventors that made them invent what they did? Say, what Rieman thought wasn't enough in Cauchy definition of integral that made him invent another. How could you summarize shortly the process which forced everyone to invent and define things differently. Cauchy , Rieman, Lebesgue , Borel, Cantor etc.

My question is devided into two parts: 1) what are the main topics of measure theory?
2) How could we create a chain process of thoughts that forced each to get a new mathematical code that thought the previous wasn't complete yet?

I'm going to use these two books as sources: Hawkins' Lebesgue's Theory of Integration and Bressoud's A Radical Approach to Lebesgue's Theory of Integration. There is nothing original here, but I am doing this as much as for myself because I am interested in the history too and as they say, the errors are all mine.

Let us start with 19th century analysis. Riemann's Theory of Integration came in 1854. It defined a Riemann-integrable function as that where the Cauchy sums approach a unique value. That is $$sum_{i=1}^n f(t_i)(x_i - x_{i-1})$$ approaches a certain limit $L$. Then came the integrability criterion given by matching the lower Darboux sums and the upper Darboux sums. $$ L = sum_{i=1}^n m_i(x_i - x_{i-1}) text{ and } U = sum_{i=1}^n M_i(x_i - x_{i-1})$$ where $m$ and $M$ are pieces over the partition. This was perceived, rightly as a great advance, and thought at that time to be the most general form of what it means to be integrable. However, there were a few problems as discovered in 19th century analysis.

Problem 1 The first problem came from Fourier analysis. Fourier analysis tried to ask how close can we get to approximating a function $f$ by a trigonometric series, and whether that series coincided with the Fourier series of $f$. Similarly in the development of the theory, we are interested in the first case of $$ sum_{i=1}^infty int f(x_i) = int sum_{i=1}^infty f(x)$$ which can be shown to not hold in the Riemann Theory of Integration.

Problem 2 Another problem is that of the Fundamental Theorem of Calculus. We are interested in showing that $$ int f'(x) dx = f(b) - f(a)$$ There were problems to the application of the FTC by the discovery of functions (by Volterra and Dini) that have bounded, nonintegrable derivatives, and thus the Fundamental Theorem fo Calculus did not apply in these cases.

Problem 3 Another issue that came up is the relation between continuity and differentiation. The first thought and intuition that people have is that a continuous function should be differentiable at most points. Weierstrauss however had an example of a function that is continuous but differentiable nowhere. We therefore want a theorem to the effect that a continuous monotone function is differentiable at most points.

Attempts were made around 1870 to put this on firm ground. Weierstrauss defined his own version of an integral using areas of rectangles. Peano attempted to describe inner content and outer content of a set $S$. Jordan defined his own measure that defined as set $S$ as Jordan-measurable if inner and outer contents match. Borel defines his general measure using the $m([a,b]) = b-a$ or the length of the measure, and he comes up with concept of countably additive $$ m (bigcup_{i=1}^infty S_k) = sum_{k=1}^infty m(S_k)$$ Of these, Borel measure is still widely used and Jordan measure too. Borel however (in modern terminology) defines his $sigma$-algebra as the $sigma$-algebra generated by intervals.

Now we come to Lebegue. First Lebesgue had to come up with a measure. He chooses the measure to satisfy the properties of 1) translation-invariance, 2) countable subadditivity, and 3) $m([0,1]) = 1$. He first however has to define Lebesgue outer-measure, which is the infimum of the lengths of the open intervals that constitute the open cover of a set $S$. He then defines the inner measure as $$m_i(S) = (b-a) - m_e([a,b]- S)$$ where it is $(b-a)$ minus the outer measure of the complement of the set $S$. When they coincide, we have the Lebesgue measure.

After we have measure, Lebesgue came up with his integral. Lebesgue's solution is to partition the "range" in the sense that Riemann integration is partitioning the "width". The integral is defined as $$sum_{i=1}^n mu(E_i) cdot f(t_i) $$ where $t_i$ is some number between $[x_{i-1}, x_i]$ in the partition of the domain and $mu$ is the measure that he defined earlier. In a way, evaluation on sets (Lebesgue measure) became a fundamental new way of looking at integration. This also however needed the introduction of Lebesgue measurable functions. (The simplest approach to get measurable functions is through something known as simple functions introduced by Riesz, and not Lebesgue.)

With this integral however, the amazingness of the Lebesgue Theory of Integration is that 1) the integral is almost fully general and 2) the integral solves all of the above problems. To see this, finish a course on measure theory and note Egorov's Theorem, Dominated Convergence Theorem, Monotone Convergence Theorem, e.t.c.