What is "uniformization"

n. The act of uniformizing.

In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if R is a subset of X × Y, where X and Y are Polish spaces, then there is a subset f of R that is a partial function from X to Y, and whose domain (in the sense of the set of all x such that f(x) exists) equals

Such a function is called a uniformizing function for R, or a uniformization of R.

To see the relationship with the axiom of choice, observe that R can be thought of as associating, to each element of X, a subset of Y. A uniformization of R then picks exactly one element from each such subset, whenever the subset is nonempty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to AC.

A pointclass $\boldsymbol{\Gamma}$ is said to have the uniformization property if every relation R in $\boldsymbol{\Gamma}$ can be uniformized by a partial function in $\boldsymbol{\Gamma}$. The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form.

It follows from ZFC alone that $\boldsymbol{\Pi}^1_1$ and $\boldsymbol{\Sigma}^1_2$ have the uniformization property. It follows from the existence of sufficient large cardinals that

• $\boldsymbol{\Pi}^1_{2n+1}$ and $\boldsymbol{\Sigma}^1_{2n+2}$ have the uniformization property for every natural number n.
• Therefore, the collection of projective sets has the uniformization property.
• Every relation in L(R) can be uniformized, but not necessarily by a function in L(R). In fact, L(R) does not have the uniformization property (equivalently, L(R) does not satisfy the axiom of uniformization).
  • (Note: it's trivial that every relation in L(R) can be uniformized in V, assuming V satisfies AC. The point is that every such relation can be uniformized in some transitive inner model of V in which AD holds.)

Uniformization may refer to:

• Uniformization (set theory), a mathematical concept in set theory
• Uniformization theorem, a mathematical result in complex analysis and differential geometry
• Uniformization (probability theory), a method to find a discrete time Markov chain analogous to a continuous time Markov chain

In probability theory, uniformization method, (also known as Jensen's method or the randomization method) is a method to compute transient solutions of finite state continuous-time Markov chains, by approximating the process by a discrete time Markov chain. The original chain is scaled by the fastest transition rate γ, so that transitions occur at the same rate in every state, hence the name uniformisation. The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero). The method was first introduced by Winfried Grassmann in 1977.