Real analysis/Riemann integration

Caveat Lector: This page is still very much in its infancy. Please expand the article as it is clearly a stub with a couple pretty formulas.

Riemann integration is the formulation of integration most people think of if they ever think about integration. It is the only type of integration considered in most calculus classes; many other forms of integration, notably Lebesgue integrals, are extensions of Riemann integrals to larger classes of functions. The Riemann integral was developed by Bernhard Riemann in 1854 and was, when invented, the first rigorous definition of integration applicable to not necessarily continuous functions.

Formally, if $x_0<x_1<\dots <x_n$, we define $N(x_0,x_1,\dots ,x_n)$ (the norm of the partition) to be the maximum of $x_1-x_0$, $x_2-x_1$, and so on up to $x_n-x_{n-1}$. Then we say the Riemann integral of a function $f(x)$ on the interval $[a,b]$ exists (or that $f$ is Riemann integrable on $[a,b]$) if and only if the limit

$\underset{a=x_0<x_1<\dots <x_n=b,N(x_0,x_1,\dots ,x_n)\to 0}{\lim }\sum _{i=0}^{n}f(x_i)(x_i-x_{i-1})$

exists. If the limit exists, and writing $\mathrm{\Delta }{x}_i=(x_i-x_{i-1})$, the Riemann integral is then

$\underset{a=x_0<x_1<\dots <x_n=b,N(x_0,x_1,\dots ,x_n)\to 0}{\lim }\sum _{i=0}^{n}f(x_i)\mathrm{\Delta }{x}_i=\underset{a}{\overset{b}{\int }}f(x)dx$.

The Riemann integral on a specified interval $[a,b]$ is a linear operator on the set of all Riemann integrable functions F such that

$F:[a,b]⟶ℝ$.

This means that, for all functions f and g with Riemann integrals over [a, b] and for all real constants $\lambda$ we have

$\underset{a}{\overset{b}{\int }}\lambda f(x)dx=\lambda \underset{a}{\overset{b}{\int }}f(x)dx$

and

$\underset{a}{\overset{b}{\int }}(f(x)+g(x))dx=\underset{a}{\overset{b}{\int }}f(x)dx+\underset{a}{\overset{b}{\int }}g(x)dx$

Lower and upper integrals and Riemann integrability

Consider a function defined in an interval [a,b] with a partition P where a=t₀<t₁<t₂<...<t_n=b, and let |P| denote max {d|d=t_i-t_i-1}.

Consider the Riemann sums

$u=\sum _{i=1}^n(\inf f([t_{i-1},t_i]))(t_i-t_{i-1})$

and

$d=\sum _{i=1}^n(\sup f([t_{i-1},t_i]))(t_i-t_{i-1})$

And consider their limits as |P| approaches 0. If they are equal, then the function is said to be Riemann integrable, and both are equal to the Riemann integral.

Proof: