How does $d\phi =d\lambda+wd\mu$ define a measure $\phi$?

Question

In Rudin's proof of the Lebesgue-Radon-Nikodym theorem, he states if $\lambda$ is a positive bounded measure and $\mu$ is a $\sigma$-finite measure, then $$d\phi =d\lambda+wd\mu$$ defines a measure, where $w\in L^1(\mu)$ and $0<w<1$.
My question is: how does that define a measure?
I am aware that $d\bar \mu=wd\mu$ implies that for all measurable, nonnegative functions $g$, $\int gd\bar\mu=\int gwd\mu$. So my first guess was to think that maybe $$d\phi =d\lambda+wd\mu$$ implies that for all measurable, nonnegative functions $g$ $$\int gd\phi =\int g(d\lambda+wd\mu)$$ But I have no idea what to do with the integral on the right. Do we distribute the integrand over the measures and integrate the sum? Do we replace $wd\mu$ with $d\bar\mu$ and suppose $d\lambda+d\bar\mu=d(\lambda+\bar\mu)$ so that at least for a characteristic function $g=\chi _E$ we get $$\int gd\phi =\int gd(\lambda+\bar\mu)=(\lambda+\bar\mu)(E)=\lambda(E)+\bar\mu(E)=\int g d\lambda+\int gwd\mu$$ I do not know what is the convention or the definition of an integral with respect to $\phi$.

Answer 1

Let's take a look at what Rudin says in his proof (it is due to von Neumann as Rudin mentions).

Here is Lemma 6.9 (note the last paragraph)

Here is the definition of the sum of two measures:

Answer 2

The idea behind the definition is Riesz theorem.

This theorem states, roughly speaking, that every measure is a linear functional and vice versa.

check the wikipedia for more info https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem

The measure $d\lambda+\omega d\mu$ is, as you mentioned, is the measure on $X$ corresponds to the linear functional $f\mapsto\int f d\lambda + \int f\omega d\mu$ by Riesz theorem.