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I thought I understood this term, but when I tried to verify this I found three different and conflicting definitions, none corresponding to mine. Is there a generally agreed definition for this term (and since it can't have all four meanings, what would be terms for the other cases) ?

Ref(1): Basic Set Theory By Nikolai Konstantinovich Vereshchagin, Alexander Shen

A mapping f: A → B has finite support if it equals the least element in B for all but a finite subset of elements of A. Requires obviously that B have some form of order and a least element.

Ref(2): Wiki: Suppose that f : X → R is a real-valued function whose domain is an arbitrary set X. The set-theoretic support of f, written supp(f), is the set of points in X where f is non-zero

Ref(3): What Does it Mean for a Function to have Finite Support?

It should mean : the function vanishes outside a set of finite measure not that only finitely many elements in the domain produce a nonzero value for the function.

(4): My own understanding

f: A → B has finite support if its domain is a finite subset of A.

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Tom Collinge
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    Reference two is normally the definition you give just before saying what it means for a function to have compact support which means it only takes non-zero values on a subset of the domain whose closure is compact. It's a common definition in topology/geometry. – Dan Rust Aug 11 '15 at 10:53
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    If you restrict reference 1 to $f\colon A \to {x\in\mathbb{R}\mid x\geq 0}$ and similarly for reference two (with the usual ordering on $\mathbb{R}$), then the two notions of support coincide. – Dan Rust Aug 11 '15 at 10:54
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    @DanielRust In that case, the support is usually taken to be the closure of the set of points where $f$ is non-zero. Otherwise the set of continuous functions with compact support is a bit too small to be interesting. – Arthur Aug 11 '15 at 10:54
  • In (4), isn't the domain of $f: A \to B$ just $A$ itself? In that case, requiring the domain of $f$ to be a finite subset of $A$ would be the same as requiring $A$ to be finite, which is probably not required for the usual notion of "finite support" to make sense. – tvk Dec 30 '15 at 18:11
  • Never use the "finite measure" definition without explaining it first. It should be considered a non-standard usage except perhaps in a specialized area. – GEdgar Apr 02 '22 at 13:00

1 Answers1

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The term "finite support" often appears in probability theory. Let me try to explain this concept in your preferred language.

Let $f:A\to B$ be a specific function and there exists a zero in $B$,

then, $f$ has finite support if $f(a)=0$ for all $a\in A\setminus X$, where $X$ is a finite set.

So your definition $(4)$ is not accurate enough, mathematically. In fact, the function's domain is still $A$.

The first three definitions mean similar things and are all correct.

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