Fubini's Theorem Holds After Changing a Condition in the Hypothesis

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Prove that the conclusion of Fubini's theorem remain in force if we replace the condition that $f$ is integrable by $\int\left(\int\left|f_x(y)\right| d \mu_2(y)\right) d \mu_1(y)<\infty$.

My solution: Simply by Tonelli's theorem, we get $\int |f| d\mu=\int\left(\int\left|f_x(y)\right| d \mu_2(y)\right) d \mu_1(y)<\infty$. Now we have the usual hypothesis for Fubini's theorem.

I doubt the solution is correct since the qustion is worth more marks in the exam.

Any help is appreciated.