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 $$ \sup_p h_p(x) \ge 1 $$ 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.