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Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N.Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct.Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study ofmulti-period mathematical finance. Later, the algebra of conditional sets wasintroduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebraof conditional sets and the concepts of conditional topology and compactness.J. Math. Anal. Appl. 437(1), 561-589 (2016)]. By means of Boolean-valued modelsand its transfer principle we show that any known result on locally convexspaces has a transcription in the frame of locally $L^0$-convex modules whichis also true, and that the formulation in conditional set theory of any theoremof classical set theory is also a theorem. We propose Boolean-valued analysisas an analytic framework for the study of multi-period problems in mathematicalfinance.

Boolean-valued models as a foundation for locally $L^0$-convex analysis and Conditional set theory