The paper consists of two sections. In the first one it is proved that any bounded non-nogative lower semi-continuous function on the unit circle is the modulus of some function with uniformly bounded Fourier sums. In the second section a simple proof of the following known result is presented: given a measurable function $f$ on the unit circle and $\varepsilon>0$, a function can be found so that $m\{f\ne g\}<\varepsilon$ and the Fourier series of ($g$ with respect to the trigonometric system and to the Walsh system converge uniformly.