Convergence of Approximate Methods
Aleksander Janicki, Aleksander Weron · 2021
In this section we collect some basic facts concerning probability measures on spaces of trajectories of continuous and cadlag processes. For omitted proofs we refer the reader to Parthasaraty (1969) and to Jacod and Shiryaev (1987) . Theorem 8.1.1 The class B C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003208877/97a026f7-5210-401c-b49b-cb2ff37c780a/content/eq1392.tif"/> of the Borel subsets of C [0,1] coincides with the smallest σ-algebra of subsets of C [0,1] with respect to which the maps π t : x → x ( t ) are measurable for all t ∈ [0,1], If μ and ν are two measures on C [0,1] then a necessary and sufficient condition that μ = ν is that μ t 1 , …, t k = ν t 1 , …, t k for all k and t 1 , t 2 , …, t k from [0,1], where μ t 1 , …, t k and ν t 1 , …, t k are measures in the k-dimensional vector space ℝ k induced by μ and ν, respectively, through the map π t 1 , …, t k : x → (x(t 1 ), …, x(t x )).