measurable linear functionals are also continuous on separable Banach spaces? It is well known continuous linear functionals are (Borel) measurable. I have read, as a remark, the converse is also true for separable Banach spaces, but I could not find any references.
 A: If "measurable" means "Borel" or even "Baire measurable", then this is true.  It's a special case of a more general result that any Borel homomorphism of Polish groups is continuous.  See for instance Kechris, Classical Descriptive Set Theory, Theorem 9.10.
Also, you can drop the word "separable".  Suppose $X$ is an arbitrary Banach space and $f : X \to \mathbb{R}$ is a Borel linear functional.  To show $f$ is continuous, it suffices to show that for any sequence $x_n \to 0$, we have $f(x_n) \to 0$.  But if we let $X_0$ be the closed linear span of $\{x_n\}$, then $X_0$ is a separable Banach space and the restriction of $f$ to $X_0$ is Borel.  So $f$ is continuous on $X_0$, meaning $f(x_n) \to 0$ as desired.
If "measurable" means "measurable with respect to a particular Borel measure $\mu$", then this is not true.  It could be that $X$ contains a dense subspace $X_0$ of full measure.  In that case, we can use Zorn's lemma to choose a linear functional $f$ which is 0 on $X_0$ and nonzero elsewhere.  Since $f = 0$ $\mu$-almost everywhere, it is $\mu$-measurable, but not continuous. 
One can also come up with examples where $f$ is $\mu$-measurable and discontinuous, and every linear functional which is $\mu$-a.e. equal to $f$ is also discontinuous.  This happens, in particular, for Gaussian measures $\mu$; see Bogachev, Gaussian Measures, Theorem 3.7.6, for an equivalent construction.  (Of course, as a consequence of measurability, there will be a Borel function $g$ with $f=g$ a.e.; but then $g$ will not be linear!)
A: You do not even need seperability. Proposition 1.2.29 in the book Barrelled Locally Convex Spaces of Bonet and Perez Carreras says that every Borel measurable linear map from a Baire locally convex space to a locally convex space is continuous. This result is due to Laurent Schwartz and it is related to his more famous Borelian graph theorem (linear maps from a separable Banach space into Souslin locally convex spaces with borelian graph are continuous).
