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.