$ S= \{(f_1,f_2)\in L^2(I)\times L^2(I)| f_1(x)+f_2(x)\leq 1, a.e.; 0\leq f_1(x)\leq a(x)\leq 1, a.e.; 0\leq f_2(x)\leq b(x)\leq 1, a.e. \}$
To make $S$ to be convex and compact, does $a,b$ need to be continuous or only $a,b \in L^2(I)$ and compatible with above constraints is enough?