I have a question regarding the construction of the $p$-local spectrum of a given spectrum $X$. I have seen in many papers people defining the dual notion of it, namely the $p$-completion of $X$, $X_p^{\wedge}:= L_{M(\mathbf{Z}/p)}X$, but never actually someone giving the precise definition/construction of the $p$-local spectrum. So I was wondering if anyone is able to give this definition/construction of the $p$-local spectrum associated to a spectrum $X$, or to provide a reference of it.
P.S. Regarding the reference; as it has been mentioned in the comments section, I am aware of the standard reference (Bousfield's paper) but I cannot understand where he defines the $p$-local spectrum construction of a spectrum $X$. Thus for a reference am looking for something more handy.