Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: | (137) |
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The setting: bounded on , a partition of , , , , , , . Finally, if we say that `` is integrable on '' and set .

**Theorem 1. ** For any two partitions ,
.

**Theorem 2. ** is integrable iff for every
there is a partition such that
.

**Theorem 3. ** If is continuous on then is
integrable on .

**Theorem 4. ** If then
(in particular, the rhs makes sense iff
the lhs does).

**Theorem 5. ** If and are integrable on
then so is , and
.

**Theorem 6. ** If is integrable on and is a
constant, then is integrable on and
.

**Theorem
. ** If on and both are
integrable on , then
.

**Theorem 7. ** If
on and is
integrable on then
.

**Theorem 8. ** If is integrable on and is
defined on by
, then is continuous on .

This class' fundamental existential dilemma / schizophrenia:

- We want to develop a mathematical world view in which
*everything*is intuitive. - At the same time, we want to
*prove*everything with no appeal whatsoever to the bad word ``intuition''.

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Dror Bar-Natan 2003-01-06