Motivation.tex

% !TeX root = Project.tex
\section{Riemann vs. Lebesgue}
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\subsection{The Motivation}
Our goal when we integrate a function is to find the amount of `space' between the graph and x-axis (or the Abscissa if you're fancy). It's pretty difficult to say how much space an arbitrary curve takes up, but we can work out the area of \dref{def:rect}[rectangles] very easily --- by calculating the {\em base $\times$ height}. Riemann integration takes advantage of this, and defines the integral of a function $f:\R \rightarrow \R$ by first bounding it from above with rectangular functions and finding the smallest of these areas (which would be an upper bound for the area of the function), then we bound the function from below and find an lower bound, then we hope that these two bounds match up. 

\begin{figure}[H] % Remember to change this to 'h'!
	\renewcommand{\subfigcapskip}{-5pt}
	\centering
	\subfigure[Bounding from Below.]{%
		\label{fig:areabelow}
		\includegraphics{Code/Area3.png}}
	\subfigure[Bounding from Above.]{%
		\label{fig:areaabove}
		\includegraphics{Code/Area4.png}}
	\caption{Approximating Area with Step Functions.}
	\label{fig:areaapprox}
\end{figure}

If the domain of our function (the `x-axis'), is $\R^2$ instead of just $\R$, we'd actually calculating a volume by finding the `space' between the graph and the axis. Curves are bad enough, but it's even harder to find the volume under a surface. Basic Riemann Integration doesn't apply to these functions --- but if we wanted, we could make a similar observation and simplification. The analog to a \dref{def:rect}[\emph{rectangle}] in $\R^3$ is a cuboid, and we can think of its size as being calculated in much the same way as in $\R^2$; the size of the base (which is now an area, instead of a length) $\times$ the height. We could continue like this for all the Euclidean spaces, $\R^n$. It'd be nice if there was a way to generalize this, and take advantage of it, so that we could integrate any function where we can make sense of \emph{`the size of the base.'} With this is mind, lets clear up some of the terms we are going to use:

\begin{description}
\item[\em Lebesgue integration\/] is the method of integration defined above. Just like Riemann Integration, it is used to find the space between the graph of a function and the domain. However, unlike Riemann, the domain can be any set at all. Thinking back to rectangles and cuboids and $base \times height$, if we wanted make sense of what it means for there to be space between the graph and the x-axis, we should be able to make sense of the `\dref{def:measure}[size]' of parts of the domain, so that we can figure out how big our base is and multiply that by the height$\ldots$
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\item[\em The Lebesgue Measure\/] seems then like it'd be a way to assign sizes to, or `\dref{def:measure}[measure]' sets in any domain; but it is not! It's a term used \emph{specifically} when we give sizes to the subsets of $\R^n$, and those sizes `\dref{def:measure}[measures]' coincided with our usual idea of the size of a set in $\R^n$ --- i.e. it is used when we are considering functions $\R^n \supset Y \rightarrow \R$, and in $\R$ the measure (length) of the interval $[1, 0]$ is 1, in $\R^2$ the measure (area) of the rectangle $[1, 0] \times [1, 0]$ is 1, in $\R^3$ the measure (volume) of the cube $[1, 0] \times [1, 0] \times [1, 0]$ is 1$\ldots$ etc. 
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\item[\bf \em The Lebesgue Integral\/] is what we get when we do Lebesgue Integration on a set with the Lebesgue Measure on it. i.e. it's Integral of normal functions $\R^n \supset Y \rightarrow \R$. This is the (basically) the same domain as the Riemann integral and so we'd want to check that the Lebesgue Integral and the Riemann Integral match up --- and then see if the Lebesgue integral is better in some way.
\end{description}

Already, we can see an improvement over Riemann Integration; Lebesgue Integration is defined on functions with any domain (kind of), and even the more restricted Lebesgue Integral can be calculated directly for any function with a domain in $\R^n$ --- so less double/triple integration nonsense. To see another benefit of the Lebesgue Integral over Riemann, let's look at the canonical example of a function for which the Riemann Integral fails; $f\colon \R \rightarrow \R,\ \ f(x) = \mathbbm{1}_\Q(x)$ (Figure \ref{fig:rational}). This is also an example of a \dref{def:sfun}[\emph{Simple Function}] since it is just the Indicator Function of a \dref{def:mablespace}[\emph{Measurable}] set, but lets not get ahead of ourselves$\ldots$ First, we should define the \dref{def:riemann}[\emph{Riemann Integral}]. 

\input{Riemann.tex}

\section{Conclusion}
\begin{figure}[ht]
	\centering
	\includegraphics{Code/Rational.png}
	\caption{An illustration of the indicator function $\mathbbm{1}_\Q(x)$, showing a few rationals that are multiples of negative powers of 2.}
	\label{fig:rational}
\end{figure}

I have run out of time :(. So, my conclusion will be me starting why the Riemann Integral fails in a certain case, how the Lebesgue Integral improves on those shortcomings, and what I would have talked about had I started earlier. 

\medskip
When we try to integrate a the indicator function of the rationals --- $\mathbbm{1}_\Q(x)$ --- we run into a problem. If we wanted to bound it from above with step functions, every possible interval we could pick will contain at least one rational. What this means is that, at a minimum, the function will be a constant one. More obviously, if we wanted to bound it from below, we'd end up with a constant zero.

\medskip
Intuition says that if we \emph{could} integrate this function, then it's integral would be very close to zero. Why? Because the set of rational numbers is so much smaller that $\R$, and so you'd expect the space underneath it to be much smaller space underneath (or really, not underneath, in this case) the reals. In fact, compared to the $\R$, the size rationals basically zero$\ldots$

\medskip
Lebesgue Integration works because it makes use of the branch of mathematics that formalizes the whole idea of what it means to find the `size' of set. Had I started sooner, I would mentioned other uses of this branch of mathematics, since being to measure things is, obviously, very useful. Namely, I would have talked about it's use in the definition of a \dref{def:rvariable}[\emph{Random Variable}].

\medskip
Nevertheless, I have included a full list of definitions as an appendix to this paper, so we did in the end (literally) do what we set out to do; work our way up to defining the Lebesgue Integral.