Example of finite measure and infinite measure

Question

Give an example of infinite measure $\nu$ and finite measure $\mu$ on reals such that $\nu ≪ \mu$, and for each $δ > 0$ there is an interval $I ⊂ R$ satisfying $\mu(I) < \delta$ and $\nu(I) ≥ 1$

My attempt:

$d\mu = f dx, d\nu = g dx$ with different positive $f, g$. This was given as a hint but I do not know how to proceed with this.

I know such an example is impossible if $\nu$ is finite from Theorem 3.5 in Folland's real analysis.

Answer 1

This can be done with scaled Lebesgue measure - take $\nu$ to be the usual measure on the line, and define $\mu$ by scaling so that

\begin{align*} \mu([0, 1]) &= 1 \\ \mu([1, 2]) &= \frac 1 2 \\ \mu([2, 3]) &= \frac 1 4 \end{align*}

and so on (and likewise for the other side of the line). Now every set with $\mu$ measure zero has $\nu$ measure $0$.

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If you prefer to deal with densities directly, take $g \equiv 1$ and $f$ any $L^1(\mathbb{R}$) function.