I'm reading the book Real Analysis of Folland, chapter 3 about signed measure, and there's some notion that confused me. In this book, he defines that $dv = fd\mu$ if $v(E) = \int_E{fd\mu}$, and that's all. And then later, he uses some notion like $d\lambda = dv - fd\mu$ and $(f - f')d\mu \ll dv$, which I think not adapt well with the definition.
I can guess that he wants to say $\lambda(E) = v(E) - \int_E{fdu}$ in the first case, and $\int_E{(f - f')d\mu} = 0$ when $v(E) = 0$ in the last one. But I need a clear vision about this notion and its property, so I don't have to guess later. It seems that this notion is really like the differential concept in Calculus, so does it has all properties differential has?
I hope someone can explain details about this notion and all its property, so I will not get confused later. Thanks.
Yes, this is what he means. Let $(X, \mathcal{S})$ be our measurable space, and let $\mathcal{M}$ denote the space of all signed measures on $X$ (e.g., $\mathcal{M} := \{ \mu:\mathcal{S} \to \mathbb{R} : \mu \text{ is countably additive} \}$; I'm ignoring the possibility that $\mu$ takes on one of $+ - \infty$ for conveniene). We have a natural embedding of $L^1(X, \mathbb{R}, \mu) \to \mathcal{M}$ for any $\mu \in \mathcal{M}$; namely, we define the measure $\mu_f(E) = \int_X f(x) 1_E d\mu$, which is again a signed measure. Now $L^1(X, \mathbb{R}, \mu)$ carries with it the structure of a vector space (it's actually a Banach space) inherited from $\mathbb{R}$; we define $ (f+g)(x) := f(x) + g(x)$ for any $f, g \in L^1(X, \mathbb{R} , \mu)$. If we'd like to turn $\mathcal{M}$ into a vector space that respects this, we'll of course define $(\mu + \nu)(E) := \mu(E) + \nu(E) $.
Incidentally, $\mathcal{M}$ comes with a natural notion of a norm that will turn it into a Banach space too.