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Is there any standard or at least "natural" measure on the set of all functions from $\mathbb{R}$ to $\mathbb{R}$? If so, what does it look like? If not, what theorems do exclude this? I have found some related threads (such as the question about probability measures on the space of all continuous functions and the question about measures on the space of measurable functions), but these are only subsets of the set of all real functions.

My guess is that any measure on the set of all functions from $\mathbb{R}$ to $\mathbb{R}$ will violate some "naturalness" conditions because the cardinality of this set is large (as discussed here). However, I am not able to flesh out this guess, and I am not quite sure which conditions should be assumed as "natural". (I know that the latter question is a soft one, but sometimes such questions do have good answers.)

I would be grateful for some references to the literature where this problem (or some results that are relevant to this problem) are discussed.

wiktoria
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    You can already flesh out this naturality issue by asking that the measure not be zero on too big sets: for example, I think the zero measure, while it always exists, is hardly "natural" in the sense you're thinking of? I assume it should also be even and translation-invariant, in the sense that $A$, $-A$, and $A + f$ where $f$ is any function should all have the same measure, the same way the Lebesgue measure on $\mathbb{R}^n$ works? Of course, these are moreso "basic" properties, but they will already eliminate many trivial or non-trivial examples you might not be thinking of. – Bruno B Jan 02 '25 at 17:49
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    I believe these answers that you point offer already very good reasons. Is it not Nedoma's pathology enough? – Augusto Santos Jan 02 '25 at 18:41
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    I guess product measures with index set $\mathbb{R}$ are standard, though not particularly natural or useful. – Michael Greinecker Jan 05 '25 at 09:03

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