A problem on almost singular measures in real analysis
Suppose that $\mu$ and $\nu$ are finite measures on a measurable space $(X,\mathcal{M})$. Prove that either $\nu \perp \mu$ or there exists $\epsilon > 0$ and a measurable set $E \subseteq X$ such that $\mu(E) > 0$ and $\nu \ge \epsilon \mu$ on $E$ (that is, every measurable set $S \subseteq E$ has $\nu(S) \ge \epsilon \mu(S)$).
![Daniel](https://matchmaticians.com/storage/user/100002/thumb/matchmaticians-7ziva3-file-1-avatar-512.jpg)
53
Answer
Answers can only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
46
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 570 views
- $7.00
Related Questions
- Use first set of data to derive a second set
- Rank, Range, Critical Values, Preimage, and Integral of Differential Forms
- Prove that $\int_0^1 \left| \frac{f''(x)}{f(x)} \right| dx \geq 4$, under the given conditions on $f(x)$
- Constructing Monotonic Sequences Converging to an Accumulation Point in a Subset of $\mathbb{R}$
- Prove that a closed subset of a compact set is compact.
- A Real Analysis question on convergence of functions
- Equality of two measures on a generated $\sigma$-algebra.
- Prove the uniqueness of a sequence using a norm inequality.