Thereom: Let $(X,S,\mu)$ be a measure space. If $\mu(X)<\infty$, $1\leq p < q <\infty$, then $L_q\subset L_p$.
But if we take $X=(0,1]$ and take the Lebesgue measure $\mu$, and define $f(x):=\frac{1}{x^{1/3}}$. Then clearly $\mu(X)=1<\infty$, and $f\in L_2$ but $f\notin L_3$, which contradicts the theorem above. But I can't see what I'm missing.