This paper consists of three parts: first a new simple proof of the semimartingale theorem of Doob is given, next the limit function is identified as the derivative of a certain $\sigma$-additive set function. Finally it is shown how the approach of Sparre-Andersen and Jessen can be generalized to give the convergence and the identification of the limit function for a semimartingale.
"On the Semimartingale Convergence Theorem." Ann. Math. Statist. 37 (3) 690 - 694, June, 1966. https://doi.org/10.1214/aoms/1177699463