Non-trivial examples of Borel measures on $\mathbb{R}$

Let $\mathcal {B}$ be the $\sigma$-algebra generated by the set of open subsets of $\mathbb{R}$. A Borel measure $\nu$ on $\mathbb{R}$ is a measure on $\mathcal {B}$ such that $\nu(K) \lt \infty$ for every compact subset $K$. The only Borel measures I know are essentially as follows.

1) Let $f$ be a non-negative $\mathcal {B}$-measurable function such that $\int_K f d\mu \lt \infty$ for every compact subset $K$ where $\mu$ is the Lebesgue measure. We write $\nu(M) = \int_M f d\mu$ for $M\in \mathcal B$. Then $\nu$ is a Borel measure.

2) Let $E$ be a countable subset of $\mathbb R$. Let $f: E \rightarrow \mathbb R$ be a non-negative function such that $\sum_{x\in K\cap E} f(x) \lt \infty$ for every compact subset $K$. We write $\nu(M) = \sum_{x \in M\cap E} f(x)$ for $M\in \mathcal B$. Then $\nu$ is a Borel measure.

I would like to know other non-trivial examples of Borel measures on $\mathbb R$.

1 Answer

Let $C$ be a cantor set. Let $f:C \to [0,1]$ be a Borel isomorphism. We put $\lambda(X)=\mu(f(X \cap C))$ for each $ X \in B(R)$. Then $\lambda$ is other non-trivial example(singular, following William Curtis remark) of Borel measure on $R$.