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2 votes
Accepted

Confused about the Radon-Nikodym theorem applied to the counting measure

The Radon-Nikodym Theorem requires the measure to be $\sigma$-finite. What you found is an example that shows that the hypothesis is crucial, as the counting measure is not $\sigma$-finite on any ...
Martin Argerami's user avatar
2 votes

For every measurable $E$ with $0<m(E)< \infty$, there is $[a,b]$ such that $m(E \cap [a,b]) > \frac{1}{2} m(E)$

In this question, we have that, if $0 < m(E) < \infty$, for any $\epsilon > 0$, there exists $M > 0$ such that $$ m(E \setminus [-M, M]) < \epsilon $$ Now, choose $\epsilon = \...
Thành Nguyễn's user avatar
1 vote
Accepted

On surface of $C = \{ x^2+y^2=1,0<z<1 \}$ there is subset A, $A_t=A \cap \{z <t \}$. Prove $\int_0^1 \lambda_2(A_t) dt= \iint_A (1-z)d \lambda_2$

Let $a_{t}={d\lambda_2(A_t)}/{dt}$ represent the 1-dimensional volume of the part of $A$ at height $t$ (defined for almost every $t\in[0,1]$). Then the left hand side is equal to $$\int_{t=0}^1\int_{z=...
Lieven's user avatar
  • 971

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