Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$
Show that $\forall a,b;c,d\in\mathbb{R},a<b,c<d.$$\int_{a}^{b}dx\int_{c}^{d}f(x+y)dy=\int_{c}^{d}dy\int_{b}^{a}f(x+y)dx.$
If $f (x+y) $ is $Riemann$ integrable on $\mathcal{R}=[a,b]\times [c,d] $,then we can easily get the equality by applying Fubini's theorem. So the key to this question is to ensure that $f (x+y) $ is Riemann integrable on $[a,b]\times [c,d] $ .Let $A= \lbrace (u,v)\in \mathcal {R^{\circ}} \quad |\quad f (x+y) \text{ is discontinuous at } (u,v)\rbrace$,we only need to prove $A$ has Lebesgue measure zero .But how can I prove $m(A)=0$ is ture?
HINT: Consider the set of discontinuities of $f$ in the interval $[a+c,b+d]$. Cover it by a countable number of intervals whose total length sums to less than $\epsilon$. For each interval $I_j$, consider the set $A_j = \{(u,v): u+v\in I_j\}$. Show that the sum of the areas of the sets $A_j$ is less than some (fixed) multiple of $\epsilon$.