关于‘for almost each object’的语法合理性及Kolmogorov复杂度相关数学含义问询

Hey there! Let's unpack your two questions clearly, since they touch on both technical terminology and English usage in academic contexts:

1. Is "for almost each object" natural and reasonable in this example context? #

In everyday conversational English, "for almost every object" would feel more idiomatic—"each" tends to emphasize individual items one by one, while "every" pairs more smoothly with "almost" to convey a general sense of "the vast majority (but not all)".

That said, in mathematical and theoretical computer science literature, "for almost each object" is a widely accepted term-specific phrasing. Researchers in the field will immediately grasp its meaning, so within this specialized academic context, it's absolutely natural and reasonable — it's just a domain-specific usage rather than a casual English one.

2. Does this phrase have a specific mathematical meaning? And is my thought that it's grammatically non-standard a mistake? #

Let's split this into two parts:

• Specific mathematical meaning: Yes, absolutely. In measure theory (which underpins fields like Kolmogorov complexity and algorithmic information theory), "for almost each" (or the more common "for almost every") is a standard technical term. It means "all objects except those in a set with measure zero".

  In the context of the Kolmogorov complexity quote:

  In particular, for almost each object it is not possible to compute even a lower bound for its Kolmogorov complexity ...

  This translates to: There exists a set of objects with measure zero (think of this as an "extremely rare" subset), and for every object not in this set, we can't even compute a lower bound for its Kolmogorov complexity. The "measure" here usually refers to the natural probability measure over objects (e.g., for binary strings, the probability of randomly generating a string in the exception set approaches 0 as the string length increases).

• Grammatical concerns: You're not missing anything! In standard everyday English, "almost each" does sound odd—"almost every" is the usual natural phrasing. But in academic writing, especially in math and CS, this is a case where technical terminology takes precedence over casual grammar norms. It's an accepted variant within the field, so it's not a "mistake" in the context it's used in.

内容的提问来源于stack exchange，提问作者MikeRoger