Uniform $L^p$ bound on finite measure implies uniform integrability
Use the inequality $$\chi\{|f|>R\}\cdot |f(x)|\cdot R^{p-1}\leqslant |f(x)|^p,$$ where $\chi(A)$ denotes the indicator function of the set $A$. Then integrate to obtain $$\int_{\{|f_n|>R\}}|f_n|\mathrm d\mu(x)\leqslant R^{1-p}\sup_k\int_X\left|f_k(x)\right|^p\mathrm d\mu(x).$$