Uniqueness of Series with Respect to the Bounded Ciesielski System

In this work some uniqueness theorems for series with respect to the bounded Ciesielski system are proved. In particular, if the partial sums Sₗ(x)=∑ₙ₌₋ₖ₊₂ˡaₙFₙ(x) of the bounded Ciesielski series ∑ₙ₌₋ₖ₊₂∞aₙFₙ(x) converge in measure to a bounded function f and ₗ|Sₗ(x)|<∞ when x∉ B , where B is some countable set, then this series is the Fourier series of the function f with respect to the bounded Ciesielski system.