Equivalence definition of relation between measures In class today, my professor gave a proof for the special case of Radon–Nikodym theorem.
Suppose $\mu,\nu$ are two finite measures on $(X,\mathcal{A})$ such that $\nu(A)\le\mu(A)$ for all $A\in\mathcal{A}$, then there exists a $\mathcal{A}$-measurable function $\triangle$ with $0\le \triangle\le 1$ such that $\int f\,d\nu=\int f\triangle\,d\mu$ for all $f\in\mathcal{M}^+(X,\mathcal{A})$.
Here $\mathcal{M}^+(X,\mathcal{A})$ means non-negative $\mathcal{A}$-measurable functions.
My professor said that "$\nu(A)\le\mu(A)$ for all $A\in\mathcal{A}$ is equivalent to say $\int f\,d\nu\le\int f\,d\mu$ for all $f\in\mathcal{M}^+(X,\mathcal{A})$". This equivalence is not so clear to me. Can anyone explain to me why this is true?
 A: One direction is obvious: for any $A\in\mathcal{A}$, the characteristic (a.k.a. indicator) function $1_A$ is an element of $\mathcal{M}^+(X,\mathcal{A})$, and so for any $A\in\mathcal{A}$, we have
$$\nu(A)=\int 1_A\,d\nu\leq\int 1_A\,d\mu=\mu(A).$$
The basic idea for the other direction is that any non-negative $\mathcal{A}$-measurable function $f$ is an limit of non-negative simple functions, and limits etc. preserve non-strict inequalities (i.e. $\leq$ and $\geq$). There's also an application of the monotone convergence theorem in there.
More precisely, first note that because
$$\nu(A)=\int 1_A\,d\nu\leq\int 1_A\,d\mu=\mu(A)$$
for all $A\in\mathcal{A}$, we can clearly see that
$$\int f\,d\nu\leq\int f\,d\mu$$
for all non-negative simple functions $f$, by the linearity of the integral. 
Now, for any $f\in\mathcal{M}^+(X,\mathcal{A})$, we have Theorem 2.10a in Folland (pg.47),



(the uniform business isn't necessary for us here). The monotone convergence theorem tells us



Thus, for any $f\in\mathcal{M}^+(X,\mathcal{A})$, we have that
$$\int f\,d\nu=\lim_{n\to\infty}\int \phi_n\,d\nu\leq\lim_{n\to\infty}\int \phi_n\,d\mu=\int f\,d\mu\\
\underbrace{\qquad \qquad}_{\substack{\text{because limits}\\\text{preserve non-strict}\\\text{inequalities}}}$$
