In Folland's real analysis textbook, there are the following propositions:
Assuming that all integrals in question exist, we have $$ (f*g)*h=f*(g*h) $$
The proof is based on the Fubini's theorem.But I don't think this meets the conditions for using the Fubini's theorem.From the conditions of the original proposition, it can only be inferred that there exists a iterated integral: $$ (f*g)*h(z)=\int \left( \int f(y)g(x-y) \, \mathrm{d}y \right)h(z-x) \, \mathrm{d}x $$ This reminds me of this famous counterexample:
$$ \int_{0}^{1} \, \mathrm{d}x \int_{0}^{1} \frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}} \, \mathrm{d}y\neq \int_{0}^{1} \, \mathrm{d}y \int_{0}^{1} \frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}} \, \mathrm{d}x $$ May I ask where I went wrong?
