I am working on this problem.
If $sup_t E|X_t|^r <\infty$ for some $r>1$, then the $X_t$’s are uniformly integrable. Note that $\mu$ is not finite.
My attempt: Using Holder's inequality, $ \frac{1}{r} + \frac{1}{s} = 1 $
$$E|X_t 1(|X|>\lambda)| ≤ (E|X_t|^r)^\frac{1}{r} (E|1(|X|>\lambda)|^s)^\frac{1}{s}; $$
$$= \ (E|X_t|^r)^\frac{1}{r} (P(X^s_t>\lambda^s))^\frac{1}{s} $$
$$ ≤\ (E|X_t|^r)^\frac{1}{r} (E|X_t|^s)^\frac{1}{s} \lambda^{-1}$$
$$= (E|X_t|^{r+s})\lambda^{-1}$$
Then
$$ E|X_t 1(|X|>\lambda)| ≤ (E|X_t|^{r+s})\lambda^{-1} $$
Then take the limit of $\lambda \rightarrow \infty $ which will give the result. However, I am not sure if I am on the right track. I would appreciate any tips.
