Re: Like a log

From: François Charton (fcharton_at_carthage.fr)
Date: 09/23/04 

Date: Thu, 23 Sep 2004 10:36:29 +0200

"Robert Israel" <israel@math.ubc.ca> a écrit dans le message de news:
>
> >This means that a function, different from k.log(x), satisfying the
equation
> >f(xy)=f(x)+f(y)
> >needs to be unbounded on *any* segment of positive real numbers...
>
> >Any idea on how such a function could be produced?
>
> Using a Hamel basis for the reals over the rationals.
> If B is such a basis and g any function from B to R, and
> log(x) = sum_{b in B} r(b) b with r(b) rational (and only finitely many
> nonzero), define
> f(x) = sum_{b in B} r(b) g(b)
>
> Then f(xy) = f(x) + f(y) for all x,y>0.
> As long as there are b_1, b_2 in B with g(b_1) b_2 <> b_1 g(b_2),
> f will not be of the form k log(x).
>

Thank you, this pretty much settles the whole thing.

Out of curiosity, do you have any idea on the weakest non-trivial
requirement that could be put on f, so that the logarithms are the only
solutions... Others posters have suggested continuous, measurable or locally
bounded, but I am under the impression that f(xy)=f(x)+f(y) for any x,y>0 =>
f(x)=k log(x) is true in much larger function spaces.

For instance, it would be true in the family of functions f defined on
positive reals such that there exists some x where f is locally bounded (a
weaker requirement than continuity, or measurability).

Francois