# "... multiple of **the** Lebesgue measure." vs "... multiple of Lebesgue measure."

I read in mathematical texts the two different versions of the same sentence, depending on the author. Which one is correct?

If a measure is translation invariant then it is a multiple of Lebesgue measure.

If a measure is translation invariant then it is a multiple of the Lebesgue measure.

Note that there is only one Lesbesgue measure.

• I've no idea what "Lebesgue measure" means, but my natural inclination would have been to suppose it's a thing, so I'd expect an article. On the other hand, most written instances don't have an article. That seems like an unlikely error, so probably the "articleless" 30% in that chart includes many people making the same "erroneous assumption" as me. Apr 26, 2016 at 12:33
• My stochastic calculus teacher would say "...multiple of Lebesgue measure." She would also say, "Here we must use the Lebesgue integral" (to distinguish it from a Leibniz/Newton integral). Apr 26, 2016 at 12:34
• ...OOPS! I should have said the "articled" 30% in my NGram above are making the same assumption as me (i.e. - the 70% that don't include an article should really be seen as even more significant, since they're going out of their way to use what superficially seems an "unusual" form). Apr 26, 2016 at 13:02
• @FumbleFingers it's not at all unusual in measure theory. "This set has measure X." Sep 25, 2016 at 2:21
• @FumbleFingers The Lebesgue measure of a set of real numbers is that set's outer Lebesgue measure if the outer measure satisfies the condition that.... Oh, never mind. Sep 25, 2016 at 2:36

In mathematical writing, both versions are acceptable, although the one without the the sounds more natural to me (especially in spoken mathematicalese). You should use the the when you want to emphasize uniqueness of (the) Lebesgue measure (for example, if you are trying to prove that another measure equals it by exploiting the properties with respect to which Lebesgue measure is unique).

There is a similar situation for Haar measure ... which reminds me of my favourite joke when giving a graduate course on measure theory.

Q: What do you call Haar measure on the two-torus? A: Lebagel measure.

Students who don't get this are more likely to fail.

I looked up my Stochastic Calculus lecture notes. It seems to be that when something (some measure, some technique) gets defined, it takes on the definite article. Also when something is contrasted with something else.

Here are some quotes from the notes (by Mingxin Xu, lecture notes Stochastic Calculus for Finance, University of North Carolina at Charlotte).

"On ([0,1], B[0,1], P), let P be the Lebesgue measure."

(definition; uses definite article)

"Some useful properties of the Lebesgue integral"

(slide title) (usage, contrast; these properties do not exist for the Riemann integral.)

For general usage, she did not include the definite article.

"In general, Expectation is defined as Lebesgue Integral"

(usage, no definite article)

In your example, the is correct. Filling in a few missing words:

If a measure [of a set] is translation invariant, then it is a multiple of the Lebesgue measure [of the set].

For the sake of clarity, I might have written this sentence as follows (although there are many possibilities):

If a measure of a set is translation invariant, then that measure is a multiple of the Lebesgue measure of the set.

Let M(set) be a measure.

If M(set) = M(set + translation), then M(set) = multiple of LM(set), where LM(set) is Lebesgue measure.

OR

If M(set) = M(set + translation), then M(set) = multiple of LM(set), where LM(set) is the Lebesgue measure.

I can't say the first version is incorrect. It's not hard to imagine practitioners adopting it conventionally. It has the virtue of brevity.

The second version appears grammatically correct, as the is specifying the Lebesgue measure as opposed to other possible measures.

Note: LM(set) is general and analogous to f(x) in the OP's comment, not to the value of f(x) at a particular value of x.

• Thanks for your answer... But i think it doesn't correspond to the sentence I asked about. There is no missing word in my sentence, apart possibly "the". I'm not talking about the measure of a set, but about a measure. The distinction is the same as between "a function f" and the evaluation of f at a value x : "f(x)". Sep 25, 2016 at 10:15
• @GillesBonnet I confused matters with my question "What is the measure of this set?" and my answers to that question. I've deleted the question and answers and added an addendum. In regard to the f(x) issue, the second part of my original answer came closer to answering your question than at first it appeared. Sep 25, 2016 at 14:12
• Even the addendum doesn't seem to capture the intended meaning of the sentence in the question. Sep 26, 2016 at 2:04
• @AndreasBlass Would you mind explaining in more detail? How does it not capture it? Sep 26, 2016 at 2:07
• The intended meaning of the original sentence is "If M is a measure and, for all sets A and all translations t, M(A) = M(A+t), then there is a single multiplier c such that, for all sets A, M(A) = c.LM(A)." You don't get the conclusion for a set A from the hypothesis for the same set A; you need the hypothesis for all sets A in order to get any conclusion at all. Sep 26, 2016 at 3:32