Let ( Ω , F , μ ) (\Omega ,\mathfrak {F},\mu ) be a measure space, μ ( Ω ) > ∞ \mu (\Omega ) > \infty . Let X n {X_n} be a sequence of measurable functions on Ω \Omega taking values in a compact metric space M M . The set of bounded stopping times τ \tau for the X n {X_n} is a directed set under the obvious ordering. The following theorem is proved: X n {X_n} converges pointwise almost everywhere if and only if the generalized sequence ∫ ϕ ( X τ ) d μ \int {\phi ({X_\tau })d\mu } converges for every continuous function ϕ \phi on M M . The martingale theorem is proved as an application.