if a sequence converges in measure in $L^p$, then converging for weak topology.

Question

Given a finite measure space $(A,\Sigma,\mu)$, for $p \in (1,\infty)$,

if {$f_n$} is a bounded sequence in $L^p(A)$ converging in measure to $f \in L^p (A)$, then {$f_n$} converges to $f$ for the weak topology on $L^P(A)$.

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Thanks in advance.