Lecture 2, September 13, 2007

To finish the preliminaries, we give an outline of the Lebesgue measure and integration theory. Be advised that the technique is a bit more involved than the outline perhaps suggested, and that this was not always mentioned - so don’t rely on these sketchy notes if your life depends on it! Read R&Y 1.3 to complete the picture, but do not try to understand all the details. The moral is that there exists a theory of integration which is built on a theory of measure and which is much more general and stronger than the theory of the Riemann integral, containing the latter as a special case. It has all the good properties which one would expect from one’s knowledge of the Riemann integral. Heuristic strategy: mentally substitute a Riemann integral of a piecewise continuous function over a finite interval whenever you encounter a Lebesgue integral. This is not mathematically true; but the risk of damage is small.

Next we introduce the concept of normed spaces and give some examples: spaces of continuous functions, finite dimensional spaces with p-norms and the L^p-spaces which one constructs in the Lebesgue theory. For the L^p-spaces one can again use the heuristic that an equivalence class of functions, which occurs in the construction, is a more-or-less continuous function on a bounded interval and that the norm is simply calculated as a Riemann integral. This is again not mathematically true, but it helps the intuition. A good reason for this: the canonical map of C[a,b] into L^p[a,b] is an injection, but not a surjection; L^p[a,b] is much larger than C[a,b]. However, if p is finite, then L^p[a,b] is simply (a model for) the completion of C[a,b] under the metric which comes from the p-norm. This follows from the density result Theorem 1.61 in R&Y, which is not at all trivial.

As a special case of the general L ^p-spaces we obtain the sequence spaces l ^p.

Read R&Y 2.1 up to and including Example 2.6.

The L ^p-spaces are very important normed spaces, but their definition is admittedly a bit technical. It seems that we are through with that now and can proceed with the general theory next time.