Uniqueness of Ciesielski series with a subsequence of partial sums converging to a bounded function

Some uniqueness theorems for series with respect to a Ciesielski system are proved. In particular, if the partial sums $S_{n_i}(x)=\sum_{n=-k+2}^{n_i}a_nf_n(x)$ of a Ciesielski series $\sum _{n=-k+2}^{\infty }a_nf_n(x)$ converge in measure to a bounded integrable function $f$ and $\sup_i|S_{n_i}(x)| \lt \infty $ when $x\notin B$, where $B$ is some countable set, with $a_n=o(\sqrt{n})$ and $\sup_i n_{i+1}/n_{i} \lt \infty $, then this series is the Fourier–Ciesielski series of $f$.