Definition of The Daniell Integral
We can then proceed to define a larger class of functions, based on our chosen elementary functions, the class, which is the family of all functions that are the limit of a nondecreasing sequence of elementary functions almost everywhere, such that the set of integrals is bounded. The integral of a function in is defined as:
It can be shown that this definition of the integral is well-defined, i.e. it does not depend on the choice of sequence .
However, the class 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 such that every function can be represented on a set of full measure as the difference, for some functions and in the class . Then the integral of a function can be defined as:
Again, it may be shown that this integral is well-defined, i.e. it does not depend on the decomposition of into and . This is the final construction of the Daniell integral.
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