cotype properties of measures

Let $K$ be a compact Hausdrauff space and $M(K)=C(K)^{*}$ the set of all bounded complex Radon measures on $K.$ Is it true that $M(K)$ is of cotype 2? I think the answer is true and to prove this its enough to check that $$\Bigg(\sum\|\mu_k\|^2\Bigg)^{\frac{1}{2}}\leq C\Bigg(\int\|\sum_k\epsilon_k(\omega)\mu_k\|^2\Bigg)^{\frac{1}{2}},$$for any measure $(\mu_k),$ which are Dirac mass. $(\epsilon_k)$ is the sequence of Rademacher functions.

• Is this another exercise? (The result is not elementary, but it is true) Commented Jul 16, 2017 at 3:13
• Also, I note that you also asked this question on MSE: math.stackexchange.com/questions/2358264/… Commented Jul 16, 2017 at 3:18
• Yemon Choi Okay! Let me explain you very clearly. This is not an excersie. I am studying Pisier's book on 'Factorization of Linear Opeartors ans Geometry of Banach space' (which was suggested to me by Pisier himself, as he is helping me and one of my collaborator on something) This fact he has used over and over again. But I really could not understood it. I am a ph.d fifth year student and on the verge of completing. So, I am really not taking any courses. There is no one in my department working on Banach space geometry. If you know the answer and can help me that would be really great. Commented Jul 16, 2017 at 4:59
• $M(K)$ is an $L_1(\mu)$ space for some measure $\mu$, but you don't even need to know that. A separable subspace $X$ of $M(K)$ is contained in a larger subspace that is isometrically isomorphic to $L_1(\mu)$ for some probability $\mu$. indeed, take a dense sequence $\nu_n$ in the unit sphere of $X$, let $\mu:= \sum_n 2^{-n} |\nu_n|$, and use the Radon-Nikodym Theorem. Commented Jul 16, 2017 at 15:02
• Oh, I just noticed that you know how to reduce your question to whether $\ell_1$ has cotype $2$. That $L_1(\mu)$ spaces have cotype $2$ is one of the first things you learn when you see the definition of cotype and is found in many books. For example, it is contained in Theorem 6.2.14 in Albiac-Kalton. It is an easy consequence of Khintchine's inequality. Commented Jul 16, 2017 at 15:39