Interchange of Expectation and integral by Fubini's theorem

Question

I read a condition that if we have $E[\int_0^1 |X_t|dt] < \infty$, then we may apply Fubini's theorem and we can have $E[\int_0^1 X_tdt] = \int_0^1 E[X_t]dt$.

Though I am not sure. Is it right? And please refer to a textbook where I can get this form of Fubini's theorem. Thanks a lot.