The Borel σ-algebra

To do probability on the real line we need a σ-algebra of subsets of \mathbb{R} to serve as our events. The obvious choice — all subsets, the power set 2^{\mathbb{R}} — turns out to be too big: there is no sensible way to assign a length to every subset (non-measurable sets get in the way). We want instead the smallest σ-algebra that still contains every set we could reasonably point at — all the intervals, and anything built from them.

That collection is the Borel σ-algebra \mathcal{B}(\mathbb{R}): the smallest σ-algebra containing all the open sets of \mathbb{R},

\mathcal{B}(\mathbb{R}) \;=\; \sigma\big(\{\, U \subseteq \mathbb{R} : U \text{ open} \,\}\big).

Its members are the Borel sets. Because it is a σ-algebra, it is closed under complement and countable unions — and so it scoops up far more than just the open sets.

A few small generators are enough

You do not need all the open sets to pin down \mathcal{B}(\mathbb{R}). Every open set of \mathbb{R} is a countable union of open intervals, so the open intervals already generate it; and each open interval is built from half-lines by the σ-algebra operations. So even the half-lines (-\infty, x] alone generate the whole thing — the fact that makes the \{X \le x\} test for a random variable work.

Watch a half-open interval and then a single point appear from nothing but half-lines, complements and intersections:

(a, b] = (-\infty, b] \cap (-\infty, a]^{c}, \qquad \{a\} = \bigcap_{n=1}^{\infty} \Big(a - \tfrac1n,\; a\Big].

The Borel σ-algebra on \mathbb{R} is generated equally well by any of the following families — each produces exactly the same \mathcal{B}(\mathbb{R}):

the open sets of \mathbb{R};
the open intervals (a, b);
the closed intervals [a, b];
the half-lines (-\infty, x].

What lives in ℬ(ℝ)

Almost everything you can name. Open sets are in by definition; their complements, the closed sets, follow at once. A single point \{a\} is closed, hence Borel, so any countable set — the integers \mathbb{Z}, the rationals \mathbb{Q} — is a countable union of points and therefore Borel too. Half-open intervals, countable intersections of open sets, countable unions of closed sets… the operations never lead out. As a rule of thumb: if you can describe a subset of \mathbb{R} explicitly, it is a Borel set.

No — and pleasingly, it is hard to write one down that isn't. A counting argument shows there are exactly 2^{\aleph_0} (continuum-many) Borel sets, but 2^{2^{\aleph_0}} subsets of \mathbb{R} in total — strictly more. So \mathcal{B}(\mathbb{R}) is a vast but proper sub-collection of the power set 2^{\mathbb{R}}.

The sets it leaves out are genuinely pathological — the classic example is a Vitali set, built with the axiom of choice, which has no consistent length at all. That is exactly why we measure on \mathcal{B}(\mathbb{R}) rather than on all of 2^{\mathbb{R}}: it is big enough for every event we care about, yet small enough to carry a sensible measure.