In Savitch's 1969 paper, "Relationships Between Nondeterministic and Deterministic Tape Complexities", he states that "all common storage functions L(n) >= lg n are measurable. In particular, any polynomial in n and lg n is measurable." His definition of measurable is: "A function L(n) is said to be measurable if there is some Turing machine with just one storage tape such that, given any input of length n, the machine will halt after a computation in which the storage tape head scans exactly L(n) squares."
So, my problem is, based on his definition, I don't understand why storage functions L(n) >= lg n would be measurable, while functions L(n) < lg n would not be. Is this somehow implicit in his definition? Or are there some publications that I should read?