Measurable Space

A measurable space is a pair $(X,\Sigma)$, where $X$ is a set and $\Sigma$ is a sigma-algebra of subsets of $X$. The members of $\Sigma$ are called measurable sets. This structure specifies which subsets can serve as the domain of a measure, but does not itself assign sizes or probabilities to them. It is a foundational object in measure theory. (math.cmu.edu)

Definition

Writing $\mathcal P(X)$ for the power set of $X$, the collection $\Sigma\subseteq\mathcal P(X)$ must satisfy:

1. $X\in\Sigma$.

2. If $A\in\Sigma$, then $X\setminus A\in\Sigma$.

3. If $A_1,A_2,\ldots\in\Sigma$, then

   $$\bigcup_{n=1}^{\infty}A_n\in\Sigma.$$

These axioms imply that the empty set is measurable and that $\Sigma$ is closed under countable intersections, finite unions and intersections, and set differences. In particular,

The closure requirements concern countable, not arbitrary, unions and intersections. (people.math.binghamton.edu)

Measurability is relative to the chosen sigma-algebra: a subset of $X$ may belong to one sigma-algebra on $X$ but not another. Thus the underlying set alone does not determine a measurable space. (math.purdue.edu)

Basic examples

Trivial structure. Every set $X$ carries the sigma-algebra

For nonempty $X$, it makes no measurable distinction between individual points.

Discrete measurable structure. At the opposite extreme, $(X,\mathcal P(X))$ makes every subset measurable.

Countable–cocountable structure. Another sigma-algebra consists of the sets $A\subseteq X$ for which either $A$ or $X\setminus A$ is countable. On an uncountable set, this can be substantially smaller than the power set. (math.cmu.edu)

Borel structure. A topological space $X$ determines the Borel sigma-algebra

Here $\sigma(\mathcal C)$ denotes the smallest sigma-algebra containing a family $\mathcal C$. On the real line, the usual Borel sigma-algebra is generated by open intervals. It is not the entire power set of $\mathbb R$. (math.cmu.edu)

Generated structures and measurable information

For any family $\mathcal C\subseteq\mathcal P(X)$, its generated sigma-algebra is

This construction is always possible because $\mathcal P(X)$ is one of the sigma-algebras in the intersection. For a single subset $A$,

with repeated members omitted. (math.purdue.edu)

A concrete example is

The structure distinguishes the two groups, but not the points within either group. More generally, when $\Sigma_1\subseteq\Sigma_2$, the second structure allows at least as many measurable distinctions as the first. This is the mathematical basis for interpreting sub-sigma-algebras as restricted information in probability theory. (math.purdue.edu)

Measurable maps

A function

is a measurable map if

The condition uses inverse images, not images: every measurable question about the output must correspond to a measurable set of inputs. If $\mathcal T=\sigma(\mathcal G)$, it suffices to verify the condition for $B\in\mathcal G$. For real-valued functions with the Borel target structure, one can test the sets $\{x:f(x)\leq a\}$ for all real $a$. (tamuz.caltech.edu)

A continuous function between topological spaces is measurable for their Borel sigma-algebras. Measurability itself, however, requires no topology or continuity. (tamuz.caltech.edu)

Subspaces and products

A subset $S\subseteq X$, whether or not it belongs to $\Sigma$, inherits the trace sigma-algebra

This makes $(S,\Sigma|_S)$ a measurable space. (mathweb.ucsd.edu)

For two measurable spaces, their Cartesian product carries the product sigma-algebra

It is the smallest sigma-algebra making both coordinate projections measurable. Infinite products are likewise equipped with the sigma-algebra generated by coordinate inverse images. These constructions provide measurable structures for collections of jointly observed variables. (mathweb.ucsd.edu)

Measures and completion

A measure space adds a countably additive function

so that for pairwise disjoint measurable sets,

Thus $(X,\Sigma)$ is the measurable space, while $(X,\Sigma,\mu)$ is the measure space. If $\mu(X)=1$, it is a probability space. Different measures can be placed on the same measurable space. (math.cmu.edu)

A measure space is complete if every subset of every measurable null set is measurable. Its completion adjoins all such subsets. Completion therefore depends on a particular measure, not merely on the measurable space. (mathweb.ucsd.edu)

For example, completing the Borel sigma-algebra on $\mathbb R$ with respect to Lebesgue measure gives the Lebesgue sigma-algebra. It contains sets that are not Borel measurable. Consequently, “Borel measurable” and “Lebesgue measurable” are distinct notions. (ocw.mit.edu)

Standard Borel spaces

A standard Borel space is a measurable space whose sigma-algebra is the Borel sigma-algebra of some Polish topology—a topology induced by a complete separable metric. The topology need not be retained as part of the measurable structure. (math.ucr.edu)

A measurable isomorphism is a bijection whose forward and inverse maps are both measurable. Every uncountable standard Borel space is measurably isomorphic to $(\mathbb R,\mathcal B(\mathbb R))$; countable standard Borel spaces have the full power-set sigma-algebra. In particular, the real line and the plane, with their usual Borel structures, are isomorphic as measurable spaces despite their different topological structures. (math.ucr.edu)

References

1. Lecture Notes on Measure Theorymath.cmu.edu
2. Math 330 – Lecture Notes: Student Edition with Proofspeople.math.binghamton.edu
3. Lecture Notes for Math 205Amath.stanford.edu
4. Lecture Notes on Probabilitytamuz.caltech.edu
5. Lecture Notes: Appendix on Standard Borel Spacesmathweb.ucsd.edu
6. Lecture Notes: Measure and Integrationocw.mit.edu
7. This Week's Finds in Mathematical Physics: Week 272math.ucr.edu