In Evan's book, the definition of Sobolev space $W^{k,p}(U)$ consists of all locally integrable function such that for each multiindex $\alpha$ with $0\leq|\alpha|\leq k$, $D^\alpha u$ exists in the weak sense and belongs to $L^p(U)$.
I am wondering why we don't we say it consists of all $L^P$ integrable functons instead of all locally integrable function. We require $D^0u \in L^p$, $u$ must be locally integrable.
