Is the pointwise supremum of a family of $L^p$ functions also $L^p$?

Question

If $T_t:L^p(X)\to L^p(X),\ t>0,$ are a family of linear operators, we define the maximal operator $T^*:L^p(X)\to L^p(X)$ by $$T^*f(x)=\sup_{t>0}|T_tf(x)|.$$

This is stated in my notes, but I'm not sure why $T^*f$ as defined necessarily lies in $L^p$ for all $f\in L^p$. Could anyone please clarify this?