Let $(X,\mathcal{A})$ be a measure space and $\mathcal{X}$ be a linear space of measurable functions from $X$ into the real numbers. So $\mathcal{X}$ could for instance be an $L_p$ space. Now in a paper that I am reading they are stating that we can now find a (topological) dual space $\mathcal{X}^*$ of finite signed measures $\mu$ on $(X, \mathcal{A})$ with $\int_X |f|d|\mu|<\infty$ for all $f\in\mathcal{X}$, such that the pairing $$ \langle \mu, f\rangle := \int_X f d\mu$$ is well defined.
1.) So this essentially means that the space of all finite signed measures has to be the topological dual of spaces of real-valued measurable functions. Is this correct? If so where can I find a proof of that?
2.) If I take the $\mathcal{X}=L_p(X, \mathcal{A}, \nu)$ for some positive measure $\nu$, then its topological dual is the space $L_q(X, \mathcal{A}, \nu)$ with $\frac{1}{p}+\frac{1}{q}=1$ and not some space of (signed) measures. So how does that fit into the above setting?
I think what you are referring to is a version of the Riesz-Markov-Kakutani representation theorem. The space of continuous compactly supported functions has these measures as their dual.
To your second question, the theorem implies that every functional given a fixed measure, is essentially integrating with a 'density' from $L^q$ when $1\leq p< \infty$. You can look at Dual Lp spaces and the references it links to. Both facts should be in any sufficiently comprehensive functional analysis book.