A counter-example to Radon-Nikodym Theorem?

measure-theory

Solution 1:

We have $$\mu([0,1]-\mathbb{Q})=0$$ but $$m([0,1]-\mathbb{Q})=1,$$ so $m\not\ll \mu$.

Related

My proof that sum of convergent sequences converges to sum of limits
Is a compact, simply-connected 3-manifold necessarily $S^3$ with $B^3$'s removed?
Calculate the infinite sum $\sum_{1}^\infty \frac{\log{n}}{2n-1}$
Confidence Interval of Information Entropy?
Ramanujan's sum related to $\tan^{-1}(e^{-\pi x/2})$
All clubs have a member among $n$ people
How to determine whether a point is inside a closed region or not?
An upper bound for Summative Fission numbers
Upper bounds for the number of intermediate subgroups
How to find $I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$
The local ring of the generic point of a prime divisor
Prove that $(\sqrt2 − 1)^n, \forall n \in \mathbb{Z^+}$ can be represented as $\sqrt{m} − \sqrt{m−1}$ for some $m \in \mathbb{Z^+}$ (no induction).