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Nov 14, 2022 at 21:52 comment added PSE Losif Pinelis and Christophe Leuridan, this question is more specific math.stackexchange.com/questions/4576471/…
Oct 28, 2022 at 20:05 comment added PSE Dear, I apologize. I'm actually interested in the case where $\lambda$ is not constant equal to one. I edited the question one more time. I hope it becomes clearer. Sorry.
Oct 28, 2022 at 8:06 comment added Christophe Leuridan @Iosif Pinellis. Yes, you are right. Although both members are continuous functions vanishing at $0$ (the function $x \mapsto \min(|x|,1)$ is assumed to be $\nu$-integrable), it is not se obvious, since we have not necessarily convergence on compact sets.
Oct 28, 2022 at 7:45 comment added Christophe Leuridan @Iosif Pinellis. You are right. In the second display, both sides are continuous functions of $x$ vanishing at $0$, but the convergence is not necessarily uniform on compact sets.
Oct 27, 2022 at 23:45 comment added Iosif Pinelis It is a bit unclear to me how you get your second display from the first one, given that $\ln$ has different branches.
Oct 27, 2022 at 21:46 history edited Christophe Leuridan CC BY-SA 4.0
deleted 18 characters in body
Oct 27, 2022 at 21:39 history edited Christophe Leuridan CC BY-SA 4.0
I corrected many errors
Oct 27, 2022 at 21:26 history edited Christophe Leuridan CC BY-SA 4.0
I added explanations
Oct 27, 2022 at 20:56 history answered Christophe Leuridan CC BY-SA 4.0