ChapterZero

Finally, some math

Reimann integration is the standard type of integration we learn in basic calculus courses– to determine the Reimann integral of a function on an interval, take a partition of that interval and take upper and lower sums of the function on that partition. Take the limit of the upper and lower sums as the refinement of the partitions approach infinity, and if these two numbers are equal, that is the Reimann integral. This integral has some shortcomings which make it less useful to anyone (mathematicians and physicists?) working in non-standard function spaces. Today, I found a nice list of them, which I would like to put here. So here goes:

• scope f is Reimann integrable on [a,b] if and only if it is continuous at almost all points of the interval– really two restrictions: integrals are only really defined on bounded intervals, to integrate over the whole number line, you need to take limits as the integral limits go to +/- infinity; the function must be mostly continuous.
• lack of completenessRemember how R is defined as the set of limit points of Cauchy sequences in Q, and has nice properties because of it? Well the function space that is Reimann integrable is incomplete in a sense analogous to Q– the integral of the limit of a set of Reimann integrable functions is not guaranteed to be equal to the limit of the integrals of that set of functions, unless the set of functions converge to their limits uniformly.

Possibly relevant posts:

• exists if continuous and B.V. (3/17/2005)
• “Good” functions (11/17/2003)
• Differentiable Flows (3/26/2005)