- Rudin : Measure algebras on abelian groups
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One also needs to specify two additional pieces of data:. For instance, Lebesgue measure theory covers the case when is a Euclidean space , is the collection of all Lebesgue measurable subsets of , and is the Lebesgue measure of. The collection has to obey a number of axioms e. Similarly, the measure has to obey a number of axioms most notably, a countable additivity axiom in order to obtain a measure and integration theory comparable to the Lebesgue theory on Euclidean spaces.
When all these axioms are satisfied, the triple is known as a measure space.
These play much the same role in abstract measure theory that metric spaces or topological spaces play in abstract point-set topology , or that vector spaces play in abstract linear algebra. One question that will not be addressed much in this current set of notes is how one actually constructs interesting examples of measures. We will discuss this issue more in later notes although one of the most powerful tools for such constructions, namely the Riesz representation theorem , will not be covered until B.
GN , math. RA Tags: boolean algebra , Loomis-Sikorski theorem , measure space , sigma-algebra , stone representation theorem , ultrafilters by Terence Tao 31 comments. A concrete Boolean algebra is a pair , where X is a set, and is a collection of subsets of X which contain the empty set , and which is closed under unions , intersections , and complements. The subset relation also gives a relation on.
Because the is concretely represented as subsets of a space X, these relations automatically obey various axioms, in particular, for any , we have:. More succinctly: is a lattice which is distributive , bounded, and complemented. We can then define an abstract Boolean algebra to be an abstract set with the specified objects, operations, and relations that obey the axioms In the literature, different authors select different initial operations and axioms when defining an abstract Boolean algebra, but they are all easily seen to be equivalent to each other.
To emphasise the abstract nature of these algebras, the symbols are often replaced with other symbols such as. Clearly, every concrete Boolean algebra is an abstract Boolean algebra. So, up to abstract isomorphism, there is really no difference between a concrete Boolean algebra and an abstract one. Now let us turn from Boolean algebras to -algebras. A concrete -algebra also known as a measurable space is a pair , where X is a set, and is a collection of subsets of X which contains and are closed under countable unions, countable intersections, and complements; thus every concrete -algebra is a concrete Boolean algebra, but not conversely.
Rudin : Measure algebras on abelian groups
As before, concrete -algebras come equipped with the structures which obey axioms , but they also come with the operations of countable union and countable intersection , which obey an additional axiom:. Any countable family of elements of has supremum and infimum. As with Boolean algebras, one can now define an abstract -algebra to be a set with the indicated objects, operations, and relations, which obeys axioms Again, every concrete -algebra is an abstract one; but is it still true that every abstract -algebra is representable as a concrete one?
As a corollary of this representation theorem, one can also represent abstract measure spaces also known as measure algebras by concrete measure spaces, , after quotienting out by null sets.
In the rest of this post, I will state and prove these representation theorems. The situation is quite different for non-commutative measure theories , such as quantum probability , in which there is basically no good representation theorem available to equate the abstract with the classically concrete, but I will not discuss these theories here. Create a free website or blog at WordPress.
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Tag Archive. One also needs to specify two additional pieces of data: A collection of subsets of that one is allowed to measure; and The measure one assigns to each measurable set. Because the is concretely represented as subsets of a space X, these relations automatically obey various axioms, in particular, for any , we have: is a partial ordering on , and A and B have join and meet.
We have the distributive laws and. As before, concrete -algebras come equipped with the structures which obey axioms , but they also come with the operations of countable union and countable intersection , which obey an additional axiom: 5. What is good mathematics? Why global regularity for Navier-Stokes is hard. Top Posts Eigenvectors from eigenvalues Almost all Collatz orbits attain almost bounded values Career advice Books Does one have to be a genius to do maths?
12 Measure Theory
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