So, what is Lebesgue integrability, and why should i care? Probably
it's relation to distributions. First comment: Lebesgue integration
interacts better with limits of sequences of functions, and as such is
important in Fourier analysis.  [ It bridges $\mathbb{N}$ and
$\mathbb{R}$, or the discrete and the continuous. ]

This brings us first to \emph{measure theory}, which discusses the
notion of \emph{length}.  We will work with the Lebesgue measure on
$\R$ and $\R^n$, which assigns a length or $n$--volume to subsets of
the line or $n$--dimensional Euclidian space.  An important concept is
a \emph{null set}, which is a set with measure zero. Countable sets
are null sets, as well as sets with dimension smaller than $n$. The
integral is built starting from the measure, following a path through
indicator functions, simple functions, non--negative functions, signed
functions and complex valued functions.

Intuitively, the difference between Riemann integration and Lebesgue
integration can be understood as chopping up the domain---evaluate a
function on a grid---versus the codomain---sum areas of contour
maps. An important property of Lebesgue integration is that integrals
of functions $f$ and $g$ which differ only over a countable set have
the same integral. In other words, $f$ and $g$ are equal \emph{almost
everywhere}.

Now, all this is again very theoretical. I still don't see why in
practice there is anything ``useful'' about Lebesgue
integration. However, it does seem to be very necessary to proove
things about convergence of series and all sorts of tricks involving
different kinds of infinities, or in other words which have to do with
$\R$ not being countable.  Supposedly it has something to do with the
difference between Hilbert spaces (energy, norm from inner product)
and Banach spaces. Important applications are probably the $1$ and
$\infty$ norms related to control and systems theory.

So, what is a Banach space? It is the generalization of completeness
of $\R$ to vector spaces, in a way that the concept of \emph{distance}
is generalized to \emph{norm}.  Completeness is imposed by requiring
that every Cauchy sequence of vectors has a limit.  So a Banach space
is a complete normed vector space.

Sequences and Cantor's diagonal argument. Some detail i got hung up
on: representation of real numbers as continued fractions, which can
be made unique, and Banach spaces of bounded sequences. For the
continued fractions, the values are taken in $\N$.  How does this
relate to the cardinality of the powerset of $\N$?


This brings us to the next question. Disentangle measure, norm,
metric, quadratic form, 2--form, inner product.