Conditional expectation Folland 3.2.17

Question

Let $(X,M,\mu)$ be a finite measure space and $N$, a sub-$\sigma$-algebra of $M$. And $\nu=\mu|_N$. If $f\in L^1(\mu)$, there exists $ g\in L^1(\nu )$, such that $\int_{E}fd\mu = \int_{E}gd\nu $ for all $E\in N$. This problem is from Folland's Real Analysis (3.2.17). In order to show the equality above we define a new measure $\lambda(E)=\int_{E}fd\mu$ then apply Radon-Nikodym to restriction of it to $N$. However, I wonder if we can compare $ \int_{}|g|d\nu$ and $ \int_{}|f|d\mu$.Is it true that the former is less than the latter, if so how can we show this?