I was given the following exercise : Prove that the Borel $\sigma-$algebra on $\mathbb R$ is also the smallest σ−algebra containing all intervals of the form $(a, b)$. Prove that it is also the smallest σ−algebra containing
all half-intervals of the form $(−∞, a]$.
My question is : what is meant by "THE" Borel $\sigma-$algebra on $\mathbb R$ ? The topology $\tau$ that defines a topological space $(\mathbb R, \tau)$ is not given right ? How can we even consider a Borel $\sigma-$algebra defined to be the smallest $\sigma-$algebra that contains $\tau$ if we don't know $\tau$ ? (More generally, I can ask myself, when does a set X have a unique Borel $\sigma-$algebra ?)
Best Answer
Given a topological space $(S,\mathcal T)$, you define the corresponding $\sigma$-algebra $\Sigma_0$ as the smallest $\sigma$-algebra on $S$ containing $\mathcal T$, that is,
$$\Sigma_0:=\bigcap_{\Sigma\text{ is $\sigma$-algebra on } S \text{ and contains $\mathcal T$}}\Sigma.$$
Note that this definition makes sense, since the intersection is non-empty (the powerset of $S$ participates), and since the intersection of $\sigma$-algebras on $S$ is again a $\sigma$-algebra on $S$.
Also note that $\Sigma_0$ is indeed the smallest (with respect to set-inclusion) $\sigma$-algebra on $S$ containing $\mathcal T$: it surely contains $\mathcal T$ (as all the $\Sigma$ do) and any other $\sigma$-algebra containing $\mathcal T$ participates in the intersection, hence is a superset of $\Sigma_0$.
In your example, $S=\mathbb R$ and $\mathcal T=\{O\subseteq R\mid \text{$O$ is Euclidean-open}\}$, where we define $O$ to be Euclidean-open whenever $x\in O$ implies that $]x-\epsilon,x+\epsilon[\subseteq O$ for some $\epsilon>0$.
Sure, we may not be able to explicitly list all the elements of $\mathcal T$ or $\Sigma_0$ (that is, give the exact "shape" of these elements), but mathematically, we are still able to uniquely and unambiguously define these sets, which is all we desire.