User:Bengtrin/sandbox

Definition of the Daniell integral

We can then proceed to define a larger class of functions, based on our chosen elementary functions, the class $L^{+}$, which is the family of all functions that are the limit of a nondecreasing sequence $h_{n}$ of nonnegative elementary functions, such that the set of integrals $Ih_{n}$ is bounded. The integral of a function $f$ in $L^{+}$ is defined as:

$If=\lim _{n\to \infty }Ih_{n}$

It can be shown that this definition of the integral is well-defined, i.e. it does not depend on the choice of sequence $h_{n}$.

Sets of measure zero may be defined in terms of elementary functions as follows. A set $Z$ which is a subset of $X$ is a set of measure zero if for any $\epsilon >0$, there exists a nondecreasing sequence of nonnegative elementary functions $h_{p}(x)$ in H such that $Ih_{p}<\epsilon$ and $\sup _{p}h_{p}(x)\geq 1$ on $Z$.

However, the class $L^{+}$ is in general not closed under subtraction and scalar multiplication by negative numbers, but we can further extend it by defining a wider class of functions $L$ such that every function $\phi (x)$ can be represented on a set of full measure as the difference $\phi =f-g$, for some functions $f$ and $g$ in the class $L^{+}$. Then the integral of a function $\phi (x)$ can be defined as:

$\int _{X}\phi (x)dx=If-Ig\,$

Again, it may be shown that this integral is well-defined, i.e. it does not depend on the decomposition of $\phi$ into $f$ and $g$. This is the final construction of the Daniell integral .