Suppose I have random variables $(X_n)_n$ that have a lower bound $$X_n > a \qquad \forall n$$ and that are finite (which I think this means that they also have an upper bound, and so $|X_n|< b$ for some $b>0$, but I am not sure about the definition of finite random variables, cause I can't find it anywhere).
Does it mean that $X_n$ are bounded in $\mathcal{L}^1$? In other words, does it mean that
$$\sup_n \mathbb{E}[|X_n|] <\infty$$
What about $\sup_n\mathbb{E}[|X_n|^p]<\infty$ for $p>1$?
Basically my question is: knowing that the random variables are bounded, do we know if their expectation is bounded?
If $|X_n| \leq M$ then $E|X_n| \leq M$ too. If $a \leq X_n \leq b$ for all $n$ then $E|X_n| \leq |a|+|b|$ for all $n$. Also, $E|X_n|^{p} \leq (|a|+|b|)^{p}$.
However saying that $X_n$ is finite does not mean that $X_n$'s are bounded. If the only hypothesis is $X_n >a$ it does no follow that $EX_n$ is bounded: take $X_n=n+a$ for a counter-example.