# Define Radon measure as an integral

Let $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be an outer Radon measure and $f \in L^1_{loc}(\mathbb R^n, \mu)$, $f \geq 0$ on $\mathbb R^n$. Now we define an outer measure $\nu: \mathbb R^n \to [0, \infty]$ by $$\nu(B) := \int_B f \; \mathrm d \mu$$ for each Borel set $B \subset \mathbb R^n$ and $$\nu(A) := \inf \{ \nu(B) \; | \; A \subset B , \, B \text{ Borel set } \}$$ for $A \subset \mathbb R^n$ arbitrary.

A proof in my book states now that $\nu$ is an outer Radon measure. But why is this the case? Is this obvious?

• What is the definition of an outer Radon measure that you're working with? – T. Eskin Mar 27 '15 at 1:20