Property C'', strong measure zero sets and subsets of the plane

Let X be a set of reals. We show that  • X has property C" of Rothberger iff for all closed F ⊆ ℝ × ℝ with vertical sections ${\displaystyle F_x}$ (x ∈ X) null, ${\displaystyle {\cup }_{x\in X}{F}_x}$ is null;  • X has strong measure zero iff for all closed F ⊆ ℝ × ℝ with all vertical sections ${\displaystyle F_x}$ (x ∈ ℝ) null, ${\displaystyle {\cup }_{x\in X}{F}_x}$ is null.