Things add up.
Measure Theory and Some Trends in Analysis (I)
December 1, 2010Posted by on
It occurred to me that measure theory, especially Lebesgue‘s theory suggests some trends of the development of modern mathematical analysis.
- Lose of Control of Objects
- Interplay between the Stage and the Actors
For the first one, I mean that, comparing to their predecessors, modern mathematicians are more willing to sacrifice some control over objects (sets, functions, etc.) in return for more useful or elegant theories. Maybe this is their surrender after being tortured for so many years by wild creatures released by themselves, but this reluctant give-up does lead to many deep results. After all, the whole point of Lebesgue‘s theory is supported by the philosophy that some small things are negligible.
Take Littlewood‘s famous three principles as an example:
There are three principles, roughly expressible in the following terms: Every set is nearly a finite sum of intervals; every function is nearly continuous; every convergent sequence of functions is nearly uniformly convergent.
For layman, these nearly and roughly things do not sound very mathematical. Anyway, mathematics is all about rigor and among all subjects of mathematics, analysis requires the most of it. It is difficult to imagine generating rigor results while you lose such an extent of control with these nearly and roughly.
However, these principles are highly appreciated and demonstrates a very deep understanding of mathematics that can only be possessed by a top-class mathematician, those with a sense of how to perceive the abstract through intuition and how to balance the requirement of rigor and the desire for beauty. Only after many years of training can one observe that without some control lost, elegance just cannot be achieved.
Similar examples are more than abundant: from convergence to convergence a.e. or in measure, from strong topology to weak topology, etc. Without these, insights will be lost. Many beautiful results would be drowned in ugly details and others would never be discovered at all.
And this is one of the trends of modern analysis: lose some control to achieve elegance and beauty.