Your question concerns these sentences:
There is no objective standard by which to measure it.
There is no objective standard to measure it.
The difference between the two sentences is that the second omits the preposition that is taken by the verb measure. This is usually either by, against, or with. The first sentence includes the preposition, and is written to avoid violating a now seldom-observed "rule" against ending a sentence with a preposition. An alternative version of the first sentence is:
There is no objective standard to measure it by.
You ask whether the infinitive must be introduced by the preposition phrase by which. The answer to that is that it need not be, but that in English this is the way we customarily talk about measuring. We measure something against, with, or by something else, such as an objective standard.
The verb measure here is transitive, and defined by the OED as:
c. trans. To bring into comparison, opposition, or competition with; to match with; to compare against.
The noun standard in this sentence conforms to the OED's 10th definition, which is:
a. (Originally fig. from 9.) An authoritative or recognized exemplar of correctness, perfection, or some definite degree of any quality.
Among the citations in support of this definition, we see:
1665 R. Boyle Occas. Refl. v. v. sig. Ll1v Men will be asham'd to be unlike those, whose Customs and Deportments pass for the Standards, by which those of other Men are to be measur'd.
Modern examples of "by which to measure" also abound.
- Bankrate presents "A gauge by which to measure your assets".
- The Albuquerque Journal tells us that the "Nation needs standard by which to measure education".
- ProjectManager.com explains that "you will have a better rule by which to measure your project" if you "extend your metric".