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Measure Theory (I): σ\sigma-algebras

The central concept of measure theory is that of a measure on a set Ω\Omega, which is a way to quantify the “size” of its subsets. However, it is usually impossible to assign such a “size” to every subset of Ω\Omega in a way consistent with the axioms of a measure. In this post, we define the notion of a σ\sigma-algebra, which we think of as the collection of all “measurable” subsets of Ω\Omega.

Preliminaries

Let Ω\Omega be a set. The power set of Ω\Omega, denoted P(Ω)\mc P(\Omega), is the collection of all subsets of Ω\Omega.

A σ\sigma-algebra (or sigma-algebra) on Ω\Omega is a set ΣP(Ω)\Sigma\subseteq\mc P(\Omega), satisfying:

  • Σ\emptyset\in\Sigma;
  • AΣ    Ω\AΣA\in\Sigma\implies\Omega\backslash A\in\Sigma;
  • A1,A2,Σ    k=1AkΣA_1,A_2,\ldots\in\Sigma\implies\bigcup_{k=1}^\infty A_k\in\Sigma.

In other words, a σ\sigma-algebra is a collection of subsets of Ω\Omega containing \emptyset, and is closed under complement and countable union. Note that closure under countable union also implies closure under finite union, by taking Ak=A_k=\emptyset for all k>Nk>N.

Here are some examples of σ\sigma-algebras.

  • Clearly, P(Ω)\mc P(\Omega) is a σ\sigma-algebra on Ω\Omega.
  • By the axioms, any σ\sigma-algebra on Ω\Omega must contain \emptyset and Ω\=Ω\Omega\backslash\emptyset=\Omega. Conversely, we can check that {,Ω}\{\emptyset,\Omega\} is a σ\sigma-algebra on Ω\Omega.
  • Let AΩA\subseteq\Omega. Then {,A,Ω\A,Ω}\{\emptyset,A,\Omega\backslash A,\Omega\} is a σ\sigma-algebra on Ω\Omega.

Example: Let Ω=4\lvert\Omega\rvert=4, and let

Σ={AΩ:A=0,2,or 4}.\Sigma=\{A\subseteq\Omega\,:\,|A|=0,\,2,\,\text{or }4\}.

Then Σ\Sigma contains \emptyset, and is closed under complement and countable disjoint union. However, the union of two 2-element subsets of Ω\Omega can contain 3 elements, so Σ\Sigma is not a σ\sigma-algebra on Ω\Omega.

Now we prove some elementary consequences of the axioms.

Proposition: For any σ\sigma-algebra Σ\Sigma on a set Ω\Omega, we have:

  • A,BΣ    A\BΣA,B\in\Sigma\implies A\backslash B\in\Sigma;
  • A1,A2,Σ    k=1AkΣA_1,A_2,\ldots\in\Sigma\implies\bigcap_{k=1}^\infty A_k\in\Sigma.

Proof: Note that

A\B=A(Ω\B),k=1Ak=Ω\k=1(Ω\Ak).  \begin{aligned} A\backslash B&=A\cap(\Omega\backslash B),\\ \bigcap_{k=1}^\infty A_k&=\Omega\backslash\bigcup_{k=1}^\infty(\Omega\backslash A_k).\qed \end{aligned}

In other words, σ\sigma-algebras are closed under countable intersection, and thus finite intersection (by taking Ak=ΩA_k=\Omega for k>Nk>N).

Proposition: If {Σα}αI\{\Sigma_\alpha\}_ {\alpha\in I} are σ\sigma-algebras on Ω\Omega, then so is αIΣα\bigcap_{\alpha\in I}\Sigma_\alpha.

Proof: Check that each of the axioms of a σ\sigma-algebra carry over to the intersection.  \qed

For any collection SP(Ω)\mc S\subseteq\mc P(\Omega) of subsets of Ω\Omega, we define the σ\sigma-algebra generated by S\mc S, denoted by σ(S)\sigma(\mc S), to be the intersection of all σ\sigma-algebras on Ω\Omega containing S\mc S.

Proposition: For any SP(Ω)\mc S\subseteq\mc P(\Omega), σ(S)\sigma(\mc S) is the smallest σ\sigma-algebra on Ω\Omega containing S\mc S.

Proof: By the previous proposition, σ(S)\sigma(\mc S) is a σ\sigma-algebra. By definition, any σ\sigma-algebra containing S\mc S must contain σ(S)\sigma(\mc S) as a subset.  \qed

This gives a new way of creating σ\sigma-algebras:

  • Let S={{ω}:ωΩ}\mc S=\{\{\omega\}\,:\,\omega\in\Omega\} be the collection of singleton subsets of Ω\Omega. Then any σ\sigma-algebra containing S\mc S must contain all countable subsets of Ω\Omega, and their complements. Conversely, the collection

    Σ={AΩ:A or Ω\A countable}\Sigma=\{A\subseteq\Omega\,:\,A\text{ or }\Omega\backslash A\text{ countable}\}

    can be checked to be a σ\sigma-algebra on Ω\Omega, so σ(S)=Σ\sigma(\mc S)=\Sigma.

  • More generally, let S={Aα:αI}\mc S=\{A_\alpha\,:\,\alpha\in I\} be a collection of disjoint subsets of Ω\Omega with union Ω\Omega. Then by a similar argument, σ(S)\sigma(\mc S) consists of all countable unions of AαA_\alpha and their complements.
  • Let Ω=Rn\Omega=\bb R^n, and let On\mc O_n be the collection of open sets in Rn\bb R^n. Then B=Bn=σ(On)\mc B=\mc B_n=\sigma(\mc O_n) contains:
    • all open and closed sets;
    • all GδG_\delta sets, ie. countable intersections of open sets;
    • all FσF_\sigma sets, ie. countable unions of closed sets;
    • all GδσG_{\delta\sigma} sets, ie. countable unions of GδG_\delta sets;
    • all FσδF_{\sigma\delta} sets, ie. countable intersections of FσF_\sigma sets;

    The elements of B\mc B are called Borel sets.

  • Let a=(a1,,an)a=(a_1,\ldots,a_n), b=(b1,,bn)b=(b_1,\ldots,b_n), where akbk-\infty\leq a_k\leq b_k\leq\infty for each k=1,,nk=1,\ldots,n. Define the half-open interval

    [a,b){(x1,,xn)Rn:xk[ak,bk) for all k=1,,n}.[a,b)\coloneqq\{(x_1,\ldots,x_n)\in\bb R^n\,:\,x_k\in[a_k,b_k)\text{ for all }k=1,\ldots,n\}.

    Let Hn\mc H_n be the collection of all half-open intervals.

    Proposition: Hn\mc H_n generates the σ\sigma-algebra of Borel sets, ie. σ(Hn)=Bn\sigma(\mc H_n)=\mc B_n.

    Proof: We give the proof for n=1n=1; the case n>1n>1 proceeds similarly.

    First, note that

    H1O1    σ(H1)σ(O1)=B1.\mc H_1\subseteq\mc O_1\implies\sigma(\mc H_1)\subseteq\sigma(\mc O_1)=\mc B_1.

    Now σ(H1)\sigma(\mc H_1) contains

    k=1[a+1k,b)=(a,b).\bigcup_{k=1}^\infty\left[a+\frac1k,b\right)=(a,b).

    Note that the topology on R\bb R is generated by a countable basis of open intervals. Hence every open set in R\bb R is a countable union of open intervals, which lies in σ(H1)\sigma(\mc H_1). Thus

    B1=σ(O1)σ(H1),\mc B_1=\sigma(\mc O_1)\subseteq\sigma(\mc H_1),

    so σ(H1)=B1\sigma(\mc H_1)=\mc B_1.  \qed

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