The distribution of a random variable in a Banach space Xwill be a probability measure on X.

A complete measure space is one in which any subset of a measure-zero set is measurable. Complete measures arise as follows (cf. Comments. For the definitions, let be a separable complete metric space and let be its Borel -algebra.

For certain aspects of the theory the linear structure of Xis irrelevant and the theory of probability

When we study limit properties of stochastic processes we will be faced with convergence of probability measures on X. Definition. If the experiment consists of just one flip of a fair coin, then the outcome is either heads or tails: $${\displaystyle \Omega =\{{\text{H}},{\text{T}}\}}$$.

So, the only reasons in support of completing the measure seem to be: 1)., it agrees with our intuition of what can be neglected, 2)., the Caratheodory extension process automatically gives a complete measure. A measure on a -algebra for which and imply for every .Here is the total variation of (for a positive measure)..

The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space.

Random measures can be defined as transition kernels or as random elements.Both definitions are equivalent. (The most common example of a separable complete metric space is ) . (viii) (E, weak) is measure compact and {0} is a Baire subset of E with respect to (E, weak).

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity.

The only reason I can think of is in the context of probability theory: using complete probability spaces forces almost-everywhere equal random variables to generate the same sigma-sub-algebra.

with a Bochner measurable function. ... A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. However, when I looked up the mathematical literature, I am unable to find a single theorem that works better for complete measures.

A probability space is a measure space with a probability measure. ).Let be a set, a -algebra of subsets of it and a positive measure on .It may happen that some set with has a subset not belonging to .It is natural, then, to define the measure on such a set as . Besides of the total variation distance which can be introduced regardless the structure of the underlying measurable space, there are other sorts of metric spaces of measures.

As a transition kernel. The space of all probability measures $\mathscr P = \mathscr M_1\cap\mathscr M_+$ is thus a closed subspace of a complete space, hence complete itself. There is a fifty percent chance of tossing heads and fifty percent for tails, so the probability measure in this example is $${\displaystyle P(\{\})=0}$$, $${\displaystyle P(\{{\text{H}}\})=0.5}$$, $${\displaystyle P(\{{\text{T}}\})=0.5}$$, $${\displaystyle P(\{{\text{H}},{\text{T}}\})=1}$$.

Every scalarly measurable function from a complete probability space into E agrees a.e.

For what reasons would I want a complete measure space?

The σ-algebra $${\displaystyle {\mathcal {F}}=2^{\Omega }}$$ contains $${\displaystyle 2^{2}=4}$$ events, namely: $${\displaystyle \{{\text{H}}\}}$$ (“heads”), $${\displaystyle \{{\text{T}}\}}$$ (“tails”), $${\displaystyle \{\}}$$ (“neither heads nor tails”), and $${\displaystyle \{{\text{H}},{\text{T}}\}}$$ (“either heads or tails”); in other words, $${\displaystyle {\mathcal {F}}=\{\{\},\{{\text{H}}\},\{{\text{T}}\},\{{\text{H}},{\text{T}}\}\}}$$.