Dual space of the intersection of locally convex vector spaces Let $S \neq \emptyset$ and let $\big((E_s,\mathcal{T}_s)\big)_{s \in S}$ be a family of locally convex vector subspaces of the same vector space. Denote by $E_s^*$ the dual space of $(E_s,\mathcal{T}_s),$ that is, the vector space of all $\mathcal{T}_s$-continuous linear functionals on $E_s.$ Furthermore, we set $M := \bigcap_{s \in S} E_s.$ 
The goal is to characterize the dual space of $(M,\mathcal{T}_M)$, where $\mathcal{T}_M := \tau(\mathcal{T}_s \cap M: \,s\in S)$ denotes the smallest topology on $M$ that contains $\mathcal{T}_s \cap M$ for every $s \in S.$
Obviously, we have 
\begin{equation}
\text{lin span}\, \bigg(\bigcup_{s\in S} E_s^*|_M\bigg) \subset M^*,
\end{equation}
where $E_s^*|_M$ denotes the set of all continuous linear functionals of $E_s^*$ restricted to $M.$
Question (general): Do we even have 
\begin{equation}
\text{lin span}\, \bigg(\bigcup_{s\in S} E_s^*|_M\bigg) = M^*\, ?
\end{equation}
Question (specific): Is something known for the dual space of the intersection of Lebesgue spaces w.r.t. $\sigma$-finite measures? For example, let $\mu$ and $\nu$ be $\sigma$-finite measures, what is known about the dual of the quotient space
$$
L_p(X,\mathcal{A},\{\mu,\nu\}) := \left(\mathcal{L}_p(X,\mathcal{A},\nu) \cap \mathcal{L}_p(X,\mathcal{A},\mu)\right) \, /\,\, [\mu+\nu],
$$
equipped with the topology $\tau(\|\cdot\|_{\mu,p},\|\cdot\|_{\nu,p})?$ Here, $[\mu+\nu]$ is the subspace of all functions $f$ such that $f = 0$ almost surely w.r.t. $\mu+\nu$ and $\mathcal{L}_p(X,\mathcal{A},\nu)$ denotes the set of all $\mathcal{A}$-measurable functions that are $p\text{th}$ power $\nu$-integrable.
 A: Let me suggest a slightly more structured situation in a first attempt at a solution: I would consider the one where we have lcs’s $E_0$ and $E_\infty$ with $E_0 \subset E_s\subset E_\infty$ and each of the inclusions continuous with dense images, in addition with the $E_s$ Banach spaces. In this case the dual of the intersection is indeed the linear hull of the dual spaces.  The question of how many of these assumptions are superfluous would require more thought.
The special case you mention fits into this scheme (assuming $p$ is finite for simplicity). One can use a suitable space of step functions as $E_0$ and the measurable functions as $E_\infty$ (in this case allowing the latter to be a non locally convex tvs).
A: $\DeclareMathOperator\span{span}$Let $M$ be a set of finite measures on $(X,\mathcal{A})$ and consider
$$
L_p(X,\mathcal{A},M) := \left(\bigcap_{\mu \in M} \mathcal{L}_p(X,\mathcal{A},\mu) \right) \, / \, [M]
$$
for some $p\in\mathopen]1,\infty\mathclose[.$ Let $q\in\mathopen]1,\infty\mathclose[$ such that $1/p + 1/q = 1$.
Then, the dual space of $(L_p(X,\mathcal{A},M),\tau(\|\cdot\|_{\mu,p} : \mu \in M))$ is given by
$$
L_p(X,\mathcal{A},M)^* = \span \left( \bigcup_{\mu\in M} \bigl\{ (\cdot,g)_\mu|_{L_p(X,\mathcal{A},M)} : g \in \mathcal{L}_q(X,\mathcal{A},\mu)\, / \, [M]\bigr\} \right).
$$
Proof (similar to Satz II.2.4 in Werner (2018) "Funktionalanalysis", Springer-Verlag): Clearly, for each $\mu \in M$ and each $g \in \mathcal{L}_q(X,\mathcal{A},\mu)$ we have that $(\cdot,g)_\mu|_{L_p(X,\mathcal{A},M)}$ is a continuous linear function on $L_p(X,\mathcal{A},M)$.
Conversely, let $f^* \in L_p(X,\mathcal{A},M)^*$ and define
$$
\nu(A) := f^*(\mathbb{1}_A) \quad A \in\mathcal{A}.
$$
Then, $\nu$ is a signed measure. Since $f^*$ is  $\tau(\|\cdot\|_{\mu,p} : \mu \in M)$-continuous there are a finite set $F \subset M$ and a positive $L > 0$ such that
$$
|f^*(g)| \leq L \max \{ \|g\|_{\mu,p} : \mu \in F\}
$$
for all $g \in L_p(X,\mathcal{A},M),$ which yields that $\nu \ll \sum_{\mu \in F} \mu$. According to the Radon–Nikodym theorem there is a density $h$ of $\nu$ w.r.t. the finite measure $\sum_{\mu \in F} \mu$, that is,
$$
f^*(g) = \int g h \, \mathrm{d} (\textstyle{\sum_{\mu \in F} \mu})
$$
for all $g \in L_{\infty}(\sum_{\mu \in F} \mu)$. Then, one can show that $h \in \mathcal{L}_{q}$ (see for example proof of Satz VIII.2.3 in Werner (2018)) and $f^* = \sum_{\mu \in F} (\cdot,h)_\mu|_{L_p(X,\mathcal{A},M)}$ on $L_p(X,\mathcal{A},M)$ since $L_{\infty}(\sum_{\mu \in F} \mu)$ is dense in $L_{p}(\sum_{\mu \in F} \mu)$.
