Let \(\left(X_{i}, \mathscr{A}_{i}, \mu_{i}ight), i=1,2\), be \(\sigma\)-finite measure spaces and \(f: X_{1} \times X_{2} ightarrow \mathbb{C}\) a measurable function. A function is negligible (w.r.t. the measure \(\mu\) ) if \(\int|f| d \mu=0\). Show that the following assertions are equivalent.

(a) \(f\) is \(\mu_{1} \times \mu_{2}\)-negligible.

(b) For \(\mu_{1}\)-almost all \(x_{1}\) the function \(f\left(x_{1}, \cdotight)\) is \(\mu_{2}\)-negligible.

(c) For \(\mu_{2}\)-almost all \(x_{2}\) the function \(f\left(\cdot, x_{2}ight)\) is \(\mu_{1}\)-negligible.