TITLE: Measures on $\mathbb{R}$ that are not translation invariant
QUESTION [5 upvotes]: I am looking for examples of measures on $\mathbb{R}$ which are not translation invariant.
The only one I could come up so far is the dirac measure.
In particular, I am looking for a measure $\mu$ on the measurable space $(\mathbb{R}, \mathcal{P}(\mathbb{R}))$ which is not invariant under translation but fulfills $\mu((a,a+1]) = 1$ for all $a \in \mathbb{R}$.

REPLY [6 votes]: Let $\omega\colon\mathbb{R}\to[0,\infty)$ be measurable. Then
$$
\mu_\omega(A)=\int_A\omega(x)\,dx
$$
defines a measure, which in general will not be translation invariant. If moreover $\omega$ is periodic if period $1$ and $\int_0^1\omega(x)\,dx=1$, then $\omega$ will satisfy your requirements.