Exercise on convergence in measure (Folland, Real Analysis)
Solution 1:
You can pass to a subsequence $f_{n_k}$ with $\int f_{n_k} \to \liminf \int f_n$ first.
This subsequence will also converge to $f$ in measure and ... then you already know what to do.
Solution 2:
You can use the Urysohn subsequence principle, but it should be modified a little bit.
(Urysohn subsequence principle). Let $x_n$ be a sequence of real numbers, and let $x$ be another real number, then $\liminf x_n\geq x$ iff every subsequence $x_{n_j}$ of $x_n$ has a further subsequence $x_{n_{j_i}}$ such that $\liminf x_{n_{j_i}}\geq x$.