Compactly supported kernels admit commuting radon integrals

Let $K_X,K_Y$ be the projections of $\mathrm{supp}F$. They are compact. If either is empty then $F=0$ and both partial and iterated integrals vanish. Otherwise choose $0\le c_X,c_Y\le 1$ in $C_c$, equal to one on these projections. Write $L_Y=\mathrm{supp}{c}_Y$. All sections are continuous and supported in the respective compact projection.

For fixed $y_0$ and $ϵ>0$, continuity of $F(x,y)-F(x,y_0)$ at each $(x,y_0)$ gives rectangles where its absolute value is $<ϵ$. Take finitely many covering $K_X$ and intersect their $y$-neighbourhoods. The resulting section difference is bounded by $ϵc_X$ everywhere, because it vanishes outside $K_X$. Thus $\mathrm{\mid }I(F_y)-I(F_{y_0})\mathrm{\mid }\le ϵI(c_X)$. This proves continuity of the $I$ partial integral (also if $I(c_X)=0$); it vanishes off $K_Y$, so has compact support. Repeating the rectangle argument with $x,y$ interchanged proves the other partial-integral assertion.

Take finitely many neighbourhoods $V_j$ with centres $y_j$ covering $L_Y$ and satisfying ${\sup }_x\mathrm{\mid }F(x,y)-F(x,y_j)\mathrm{\mid }<ϵ$ on $V_j$. A partition ${\psi }_j$ subordinate to them sums to one on $L_Y$. Put $H(x,y)=c_Y(y){\sum }_j{\psi }_j(y)F(x,y_j)$. Since $F=c_{Y}F$, on $L_Y$ the convex-combination bound gives $\mathrm{\mid }F-H\mathrm{\mid }\le ϵc_X{c}_Y$; off $L_Y$ both vanish. This also holds off $K_X$. Each tensor factor is in the appropriate $C_c$.

Linearity gives $I_x{J}_{y}H={\sum }_{j}I(F_{y_j})J(c_Y{\psi }_j)=J_y{I}_{x}H$. Applying positivity twice to the error bound in either order yields $\mathrm{\mid }{I}_x{J}_{y}F-J_y{I}_{x}F\mathrm{\mid }\le 2ϵI(c_X)J(c_Y)$. The cutoff integrals are finite real numbers, so arbitrariness of $ϵ$ proves equality, including either zero cutoff integral. Real and imaginary parts prove the complex claim.