Summability of series with respect to a Haar system by the $(C,1)$ method

For a Haar-system series we prove that if the lower bound of the $(C,1)$ means of the series is larger than $-\infty$ on a set $E$ of positive measure, then the series converges to a finite function almost everywhere on $E$; from this it follows that Haar-system series are not summable by the $(C,1)$ method to $+\infty$ on sets of positive measure.