Constructing Measure

Question

Let $f:\mathbb R \rightarrow \mathbb R_0^+ \text{ be an integrable function}$

a) $\phi(Y) =\int_Yfd \lambda$

$Y \text{ is an element of the borel-}\sigma-\text algebra$
$\lambda \text{ is the lebesuge measure}$

I already proofed that $\phi$ is a well defined measure. Now I need to show that there exists a measure $\phi$ for the borel-$\sigma-algebra$ with the following properties.

1. $\theta(\mathbb R)=1$

2. $N\subset \mathbb R \text{ is a zero set exactly when }\text{ N is a zero set for } \lambda$