# Thread: Measuring the reals

1. ## Measuring the reals

Let $\displaystyle (\mathbb{R}, \Sigma, \mu)$ be a measure space, $\displaystyle \mu$ some measure. There exists at least one Lebesgue integrable function. Then does it hold that,

$\displaystyle \mu(\mathbb{R}) < \infty \Rightarrow \mu(\mathbb{R}) = 0$?

2. What exactly are you asking? As for the last part you can just take $\displaystyle \mu$ to be the zero measure (not interesting but a measure none the less).

3. Originally Posted by putnam120
What exactly are you asking? As for the last part you can just take $\displaystyle \mu$ to be the zero measure (not interesting but a measure none the less).
It's okay - I found an example (although I can't remember what it is...).

Basically, my question was: does there exist a (positive) measure such that $\displaystyle \mu(\mathbb{R}) < \infty$ but which is not the zero measure.

4. Hello,

Well just make $\displaystyle \mu$ a Dirac measure

5. Let $\displaystyle \nu$ be any measure defined on $\displaystyle \mathbb{R}$ such that there exists $\displaystyle A\in \Sigma$ with $\displaystyle 0<\nu(A)<\infty$. Then the measure $\displaystyle \mu(E)=\nu(E\cap A)$ will be finite.