Things add up.
Measure Theory and Some Trends in Modern Analysis (II)
December 3, 2010Posted by on
The second trend is even more obvious in measure theory, namely, the interplay between stages and actors.
By stages, I mean number systems, or, in general, sets; and actors refer to functions and transformations.
In classical analysis, these two are separated and handled quite differently in nature. Sets are like sites on which one constructs things and functions are those buildings constructed, though closely related to each other, these two are distinct and they do not interact.
However, in measure theory, things are different. These two act intimately with each other and this interaction is quite substantial in the theory in the way that it ties functions closely to the topology or other properties of the underlying space.
On the one hand, many functions are defined according to properties of sets/ spaces while others are constructed just to demonstrate something about the space, among which characteristic functions serve as good examples.
On the other hand, sets are manipulated with respect to certain functions. As an example, consider the trick when the whole space is partitioned into preimages of functions. Another good example comes from Riesz Representation Theorem for locally compact Hausdorff spaces and positive linear functionals, where the interplay of stages and actors is so apparent and plays such an central role.
This phenomenon somehow reminds me of Einstein’s Special Relativity, which states that time-space is not something constant, but keeps changing, due to the energy and matter inside.
Also, the idea that given any measure and any function, a new measure can be given by just taking their product also resembles principles in Special Relativity that length, area, or volume are subject to change within different frame of reference. But this is beyond the scale of this post.