measure zero: meaning, definition, pronunciation and examples
C2Academic/Technical
Quick answer
What does “measure zero” mean?
A mathematical concept describing a set so small that it has no length, area, or volume, despite potentially containing infinitely many points.
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical concept describing a set so small that it has no length, area, or volume, despite potentially containing infinitely many points.
Used metaphorically to describe something so insignificant or negligible that it can be effectively ignored in practical terms.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. The term is identical in both mathematical communities.
Connotations
Purely technical with no regional connotations.
Frequency
Equally rare outside specialized mathematics in both regions.
Grammar
How to Use “measure zero” in a Sentence
[Set/Property] + [has/is of] + measure zeroVocabulary
Collocations
Examples
Examples of “measure zero” in a Sentence
noun
British English
- The Cantor set is a classic example of an uncountable set with measure zero.
- One must show the set of discontinuities is of measure zero.
American English
- The function is differentiable almost everywhere, meaning except on a set of measure zero.
- In probability, an event with measure zero is said to occur 'almost never'.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core term in real analysis, measure theory, probability, and dynamical systems.
Everyday
Extremely rare; only in metaphorical extension by highly educated speakers.
Technical
Precise meaning in mathematics and theoretical physics.
Vocabulary
Synonyms of “measure zero”
Strong
Neutral
Weak
Vocabulary
Antonyms of “measure zero”
Watch out
Common Mistakes When Using “measure zero”
- Using 'measure zero' to mean 'no measure' or 'unmeasurable'. (It is a specific measure, namely zero).
- Confusing with 'empty set'. (A set can be non-empty but still have measure zero, like the set of rational numbers between 0 and 1).
FAQ
Frequently Asked Questions
No. An empty set has measure zero, but many non-empty sets (like the set of rational numbers) also have measure zero.
Yes. For example, a single point has zero length (measure zero) on a line, but a line segment has positive length on a line and zero area (measure zero) in a plane.
Primarily yes, but it can be used metaphorically in academic or intellectual discourse to denote something utterly insignificant.
They are closely related. A property holds 'almost everywhere' if the set of points where it fails has measure zero.
A mathematical concept describing a set so small that it has no length, area, or volume, despite potentially containing infinitely many points.
Measure zero is usually academic/technical in register.
Measure zero: in British English it is pronounced /ˈmeʒə ˈzɪərəʊ/, and in American English it is pronounced /ˈmɛʒər ˈziroʊ/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “For almost all x (meaning: for all x except a set of measure zero)”
Learning
Memory Aids
Mnemonic
Think of drawing a line with a pencil: the line has length, but the individual graphite points making the line have zero length collectively. Those points are 'measure zero' on the line.
Conceptual Metaphor
ABSTRACT INSIGNIFICANCE IS ZERO SIZE (e.g., 'His influence on the decision was of measure zero').
Practice
Quiz
Which of the following sets has Lebesgue measure zero on the real line?