What is the first moment of a measure?

Question

I know what the first moment of a random variable is, however the paper I am reading refers to measures with "finite first moment", and I am unable to find a definition for this.

Thanks.

Answer 1

If $\mu$ is a measure on $(\Omega,\sigma)$ then moments are defined:

$$\int_\Omega x^n d\mu(x)$$

Answer 2

Actually even when the function $x^n$ is well-defined, the value of $\int_{\Omega} x^n d\mu(x)$ is not always finite. Then if this intergal is finite, it is understood as $n$th moment of the measure $\mu$.