Borel set

In plain terms

Not every collection of points can be assigned a size, and this genuinely shocked mathematics when it surfaced. If you insist that every set has a volume you get contradictions, most famously a solid ball that can be cut into five pieces and reassembled into two balls each the same size as the original.

The repair is to be selective. Fix a family of well-behaved sets, starting from simple ones like intervals, and allow yourself to combine them in countably many steps, so you take the union of a sequence of sets, take the complement of a set, and repeat, and you only ever measure sets built this way. That family is the Borel sets.

The restriction sounds drastic and costs nothing, as every region you could describe, draw or physically distinguish is a Borel set. The pathological sets need the axiom of choice, a principle allowing selections no rule could specify, so they can be proved to exist but never exhibited, and probability theory lives happily inside the fence.

In CSD

Every measure in the corpus is a Borel measure, whether on the state space or on the group of unitary transformations, and every outcome region whose volume becomes a Born weight is a Borel set. This isn't decoration, as the programme's central claim is a statement about the measures of regions, and it would be meaningless for regions that have no measure.

One detail the constructions lean on is that the boundaries between outcome regions are null sets, meaning sets of volume zero. A state sitting exactly on a boundary is ambiguous about which outcome it belongs to, but the ambiguity has probability zero, so no prediction ever depends on resolving it.

Mathematically

The Borel sigma-algebra of a topological space is the sigma-algebra generated by its open sets, the smallest family that contains the open sets and is closed under complements and countable unions.

The corpus's measures are Borel probability measures on the projective space CP^(N-1) and on the unitary group U(N), and the outcome regions are Borel sets with Fubini-Study-null boundaries. The null-boundary fact is used in the proofs and not just noted.

The name

Emile Borel (1871-1956) was born at Saint-Affrique in the Aveyron, a pastor's son, and topped the entrance examinations for both the Ecole Normale Superieure and the Ecole Polytechnique, choosing the Normale to the Polytechnique's lasting chagrin. He taught at Lille and then in Paris, at the Normale and the Sorbonne, and founded measure theory around 1898.

His second life was public, as a Radical-Socialist deputy in the Chamber, Minister of the Navy in 1925, and, at seventy, a Resistance activist, which earned him a spell in Vichy detention and, after the war, the Resistance Medal. The infinite monkey theorem is his image, from a 1913 paper. Few mathematicians have run a ministry, and fewer still after founding a field.

Related entries

• Outcome regionAn outcome is a region, and the apparatus alone draws its boundary, so the preparation decides where the trajectory lands and doesn't get to draw the map.
• Epistemic measureCondition on what a preparation actually fixes, and the ignorance left over is a measure, and that measure is a theorem rather than a stipulation, as the arena's own volume disintegrates into exactly it.
• Liouville measureA deterministic substrate needs a notion of volume that its own motion can't distort, and on CSD's sector that volume is forced and it's the Born measure.
• Euclidean spaceOrdinary geometry, extended to complex numbers, is the stage on which every finite quantum system in the corpus stands.
• Typicality volumeAlmost every way the experiment could have started gives the quantum statistics, and that isn't an assumption about probability, as it's a measured volume.