How much measure theory needed for advanced functional analysis?

Question

I have done a basic course in Functional Analysis following Kreyszig, and will start another one on more advanced topics that uses Rudin's Functional Analysis text. However, I have never studied Measure Theory and Lebesgue Integration. How much of it is needed for Functional Analysis? Kreyszig did not use any, and Rudin's preface mentions that it is needed, but can it be picked up as I go along? Or should I defer this course and study Measure Theory first?

Answer 1

Certainly it is good to have an idea about the issues addressed by modern theories of measure and integration.

Probably not so intensely relevant to most peoples' interest in functional analysis is general measure theory. Historically, this arose in part from the natural idea of attaching a sense of "measure" to sets on the line, the plane, 3-space, and so on, extending the notions of length, area, etc. This was also entangled with set-theoretic and syntactical-logic issues about describing/classifying subsets of the line, the plane, and so on.

Perhaps those issues turned out to be barely tractable, and in some cases depending on the Axiom of Choice and the Continuum Hypothesis... so not resolvable in "primitive/naturalist" terms. But, ok, perhaps we're not interested in that aspect of "measure theory".

That is, maybe we simply want to be able to integrate functions that may not be nice enough to integrate by the Riemann idea. BUT apart from literal constructions, what we truly want is some construction that will produce structurally similar or identical outcomes/properties. So it's not the construction, but the properties.

In fact, even prior to any discussion of measure-and-integral, people probably know what properties they need/expect/require of "integrals". Linearity... and, more trickily, some vaguely specified continuity properties. The basis continuity properties for Riemann integrals are often not visible... so one of the chief virtues of "Lebesgue integrals" is that the development makes explicit the properties that we want/need (and gives hypotheses under which these hold, even for non-elementary functions).

So, for many purposes, if you accept "integration of not-necessarily-continuous..." functions as a black box, but with clear properties, you don't need to know the "internals" of the construction of the "integral".