Remark 11.1.1 (General Plan).label

1. (1)

   General Measure Theory

   1. (a)

      Caratheodory Construction

   
   
2. (2)

   Integration

   1. (a)

      Measurable Sets and Functions

   2. (b)

      Fatou Lemma

   3. (c)

      Monotone/Dominated Convergence

   
   
3. (3)

   Radon-Nikodym Derivative

Let $E\suf \R^{n}$ be a bounded domain and suppose $\pd\cl{E}$ is piecewise smooth, then $E$ is regular and the volume of $E$ is given by

interpreted as a Riemann Integral. The extension of $\text{vol}(E)=\abs{E}$ to Lebesgue measurable sets is denoted by $m(E)$ called the Lebesgue measure of $E$ with several properties:

1. (1)

   If $E$ is regular then $\abs{E}=m(E)$

2. (2)

   $E\supf E'\implies m(E)\geq m(E')$

3. (3)

   Countable Additivity

The set of measurable functions is closed under taking limits and for $E\suf\R^{n}$ regular, $f\in C^{0}(\cl{E})$ then $\integral{E}{}fdm=\integral{E}{}f(x_{1},...,x_{n})dx_{1}\cds dx_{n}$ Let $P=\curl{x_i}_{0}^{n}$ be partition of $[a,b]$ then $a=x_{0}<x_{1}<\cds<x_{n}=b$. Let $f:[a,b]\to\R$ be bounded then the lower and upper Riemann Sums are given by

and for any refinement $P'=P\cup \curl{y_1,...,y_m}\supf P$ we have

By monotonicity $L(f)=\define \SUP{P}L(f,P),U(f)\define\INF{P}U(f,P)$ exists. Given $f:[a,b]\to \R$ bounded we say that $f$ is Riemann Integrable and write $f\in RI([a,b])$ if $L(f)=U(f)$. If $f\in C^{0}*([a,b])$ then $f\in RI([a,b])$. Indeed for any $\e>0$ there exists a partition such that $\abs{U(f,P_\e)-L(f,P_\e)}<\e$ hence by taking limit $\abs{U(f)-L(f)}<\e$ and since $\e>0$ was arbitrary, $U(f)=L(f)$. ^[1]