# 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.

• You are talking about linear functionals? – Michael Greinecker Dec 6 '16 at 14:12
• @MichaelGreinecker Yes. Sorry for being imprecise. – newbie Dec 6 '16 at 14:15
• Your question provides rather little information.. Is it Lusin's theorem what you are looking for? – Hannes Dec 6 '16 at 14:15
• @Hannes I am asking if all measurable linear functionals on separable Banach spaces are continuous. It is not the exact statement of Lusin. – newbie Dec 6 '16 at 14:17

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!)