Remark. Topology, Borel sets and coordinate observables [fdip-004C]

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On the four-cycle, a configuration is a point \(x=(x_0,x_1,x_2,x_3)\in \mathbb R^4\). The product topology makes \((-1,1)^4\) open. The Borel sigma-algebra also contains the closed half-space \(H=\{x:x_0\le 0\}\), because its complement is open. Thus Borel sets include both open and closed coordinate conditions.

The coordinate observable \(p_0(x)=x_0\) is continuous: the preimage of an open interval is its coordinate cylinder. For every Borel \(A\subseteq \mathbb R\), the set \(p_0^{-1}(A)\) is Borel by the coordinate-product definition and Proposition [fdip-002E], proving measurability. The indicator \(\mathbf 1_H\) is also measurable: its preimage of a Borel set is one of \(\varnothing ,H,H^c,\mathbb R^4\). It is not continuous at the origin, since its values on \((1/k,0,0,0)\), for positive integers \(k\), are all zero while its value at their limit is one.

Mathlib's continuous_apply and measurable_pi_apply give the coordinate projection facts for product topologies and product measurable spaces, respectively.

These deductions distinguish topology, measurable sets and measurable maps using the conventions of [tao2011measure, Definitions 1.4.16 and 1.4.32, pp. 87 and 95, preliminary version]. They say nothing by themselves about integrability. For example, the constant observable \(1\) is measurable, but its integral against the coordinate reference measure on \(\mathbb R^4\) is infinite.