Measure Theory from Three Primitives

We derive the core structures of classical measure theory - σ-algebras, measures,measurable functions, the Lebesgue integral, and the almost-everywhere qualier from the single primitive of distinguishability, expressed through the Tree of ContinuaC and its tolerance filltration structure. No axioms beyond the three primitives (same,dierent, opposite) are invoked. The σ-algebra axioms, the Carathéodory extension,and the construction of the Lebesgue integral via simple functions and monotonelimits are shown to be scaolding erected to access a structure that is native to C.Measure theory is the theory of same and dierent applied to uncountable spaces.The primitive opposite (chirality) plays no role: measure theory lives entirely in theχ = 0 sector of Z(α, χ, ε).