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Let $X = C(\mathbb R)$ be the Fréchet space of real-valued continuous functions. For each $f \in X$ and each compact set $D \subseteq \mathbb R$, let $$[f]_D = \{ g \in X : \mbox{$g(t) = f(t)$ for all $t \in D$} \} $$ be the equivalence class of functions which agree with $f$ on the set $D$.

I refer to this object frequently in a paper I am writing. The notation $[f]_D$ is short and sweet, but the terminology "equivalence class of functions" is too clunky.

Question: Is there a better name for the object $[f]_D$?

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You might call $[f]_{D}$ the set of all functions $D$-equivalent to $f$. Through some abuse of terminology, you might also refer to $[f]_{D}$ as "$f$ mod $D$". –  Adam Bjorndahl Aug 26 '11 at 18:19
It would help if you said more about how you want to use it or why you need a different name. It might be useful to choose several names, depending on whether you are picking an representative from Pete, acting via an operator on Alice, merging Mary and Mark using a higher equivalence relation. When I am stuck for a name to use temporarily, I revert to Fred. The pizza parlor people appreciate it. Gerhard "Starbucks Baristas Also Like It" Paseman, 2011.08.26 –  Gerhard Paseman Aug 26 '11 at 18:23
Here's a representative paragraph, Gerhard: "For every $f \in X$, the measure $P_D(f, \cdot)$ is supported on the equivalence class of functions $[f]_D$. This means that $P_D(f, [f]_D) = 1$, so for $P_D(f,\cdot)$-almost every $g$, $g(t) = f(t)$ for all $t \in D$. For any $g \in [f]_D$, the measures $P_D(f,\cdot)$ and $P_D(g,\cdot)$ are equal." The addition of this paragraph now warrants the pr.probability tag. –  Tom LaGatta Aug 26 '11 at 18:29
Also, if you have a notation for the restriction operator to $D$, say $\rho_D:C(\mathbb{R})\to C(D)$, then $[f]_D=\rho_D^{-1}(f_{|D})=f+\operatorname{ker} \rho_D$. –  Pietro Majer Aug 27 '11 at 6:55
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4 Answers

What about "an extension class" or "an extension family" (from D)?

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actually, since the equivalence involved is "having the same restriction on $D$", also "restriction equivalence class", "restriction class", "D-restriction class" should make sense. –  Pietro Majer Aug 27 '11 at 7:05
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How about "$D$-class"?

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For ease of exposition, it is often good to supress those parameters that do not vary often. In the fragment suggested by Tom LaGatta in the comments, there is enough explanatory information and samples of forms of usage that I am quite comfortable with it as a first, second, or even third paragraph. I, too, would get tired of writing or reading 10 such paragraphs. So I would set up a context paragraph and occasional reminder sentences, such as follows (forgive the lack of Tex):

"Since D won't change much in the following, we will write [f] sometimes to hide the dependence on D. Also, we will use the nickname 'class' sometimes to refer to the fact that [f]_D is an equivalence class of functions."

And then occasionally

" so we have that expression(h) does this for us (remember [h] is short for the class [h]_D)..."

I hope this helps.

Gerhard "Ask Me About System Design" Paseman, 2011.08.26

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Thanks, Gerhard. It's not apparent from the paragraph I showed you, but we actually need to change the parameter quite frequently. The central theorem of that section is that the conditional measure $P_D(f,\cdot)$ varies weakly continuously in both parameters $D$ and $f$. –  Tom LaGatta Aug 26 '11 at 19:57
Do what you think is best of course. In my opinion, you can still drop parameters briefly even if they are used often. E.g. 'For ease of reading, understand P is a shorthand: for the next section, P(...)=P_D(f,...) and Q(...)=P_C(g,...)'. It takes more care than skill to set up a shorthand for the reader to follow, as long as the shorthand is clear and consistent in its usage. Gerhard "Ask Me About Consistent Usage" Paseman, 2011.08.26 –  Gerhard Paseman Aug 26 '11 at 21:08
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Why don't you just say that for any function $\phi\in C(D)$ there is a measure $P_\phi$ on the class $[\phi]$ of functions $f\in C(\mathbb R)$ whose restriction $f|_D$ coincides with $\phi$?

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Thanks, R W. We do everything on the Fréchet space $C(\mathbb R)$, and vary the compact sets $D$. It is much easier to look at a single $f \in C(\mathbb R)$, restrict it to $D$, then vary the set $D$ and function $f$. Otherwise, it would be hard to compare functions $\phi \in C(D)$ and $\phi' \in C(D')$ for nearby $D$ and $D'$. –  Tom LaGatta Aug 26 '11 at 19:59
With the notation I suggest you can still talk about $P_{f|_D}$, but of course it's up to you :) –  R W Aug 27 '11 at 1:43
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