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We propose a sequential topology on the space of sub-$\sigma$-algebras of aseparable probability space $(\Omega,\mathcal{F},\mathbb{P})$ by linkingconditional expectations on $L^{2}$ along sequences of sub-$\sigma$-algebras.The varying index of measurability is captured by a bundle space construction.As a consequence, we establish the compactness of the space ofsub-$\sigma$-algebras. The proposed topology preserves independence and iscompatible with join and meet operations.

A compact topology for $\sigma$-algebra convergence