Absolute convergence of the double fourier–franklin series

We prove that, for every $0<\epsilon<1$, there exists a measurable set $E\subset{T=[0},1]^{2}$ with measure $|E|>1-\epsilon$ such that, for all $f\in L^{1}({T})$ and $0<\eta<1$, we can find $\tilde f \in L^{1}({T})$ with $\iint\nolimits_{T}| f(x,y)-\tilde f (x,y)| dxdy\leq\eta$ coinciding with $f(x,y)$ on $E$ whose double Fourier–Franklin series converges absolutely to $f$ almost everywhere on $T$.