An extension of the Martingale Convergence Theorem to splines

We prove the analogue of  the Martingale Convergence Theorem for polynomial spline sequences.

Given a natural number $k$ and a sequence $(t_i)$ of knots in $[0,1]$ with multiplicity $\leq k-1$, we let $P_n $ be the orthogonal projection onto the space of spline polynomials in $[0,1] $ of degree $k-1$, corresponding to the grid $(t_i)_{i=1}^n$. Let $X$ be a Banach space with the Radon-Nikod\'{y}m property. Let  $(g_n)$ be a bounded sequence in the Bochner-Lebesgue space $L^1_X [0,1]$ satisfying 

$g_n = P_n ( g_{n+1} ).$

We prove the  existence of $\lim_{n\to \infty} g_n(t) $ in $X$,  for almost every $t \in [0,1].$