Timeline for Does convergence of a sequence of subharmonic functions imply the vague convergence of their Riesz measures?
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| Oct 3, 2020 at 21:00 | comment | added | M. Rahmat | Sorry, but I am confused! I have two problems. 1) How did you get the measures of $K$? 2) According to my understanding, the Riesz measure $\mu_n$ of $u_n$ is defined as $$\int_D \Psi d\mu_n =a\int_Du_n\Delta \Psi d\lambda$$ for all test function $\Psi$ in $D$ ($C^\infty $-smooth compactly supported in $D$). Now if you take for $D$ the ball of, say, radius 2 centered at the origin, then $u_n$'s are bounded and by dominated convergence theorem we get, as $n\to\infty$, $$\int_Du_0 \Psi d\lambda=\int_D \Psi d\mu_0=\int_D \Psi d\nu.$$ Isn't this a contradiction with what you are saying? | |
| Oct 3, 2020 at 13:05 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |