Savitch's use of measurability
I think that this definition is nowadays known under the term space-constructible function. There are functions in sublogarithmic space that are space constructible, while others are not. 

http://dl2.acm.org/citation.cfm?id=31171 Andrzej Szepietowski: There are no fully space constructible functions between log log n and log n. Information Processing Letters 24(6),361-362.