Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the No-Arbitrage hypothesis and let $M$ denote the set of equivalent martingale measure. Note that each $\mathbb{Q} \in M$ can be viewed as a function that assigns a non-negative value $\mathbb{Q}[A]$ to each $A \subset \Omega$.
(a) Show that if $\mathbb{Q}_0$,$\mathbb{Q}_1 \in M$, then for any $0 \leq t \leq 1$, the probability measure $\mathbb{Q}_t$ defined by $$\mathbb{Q}_t[A] = (1-t) \mathbb{Q}_0[A]+ t\mathbb{Q}_1[A]$$ is also in $M$.
(b) Let $C$ be a contingent claim and $\Pi(C)$ denote the set of all arbitrage-free prices of $C$. Prove that $\Pi(C)$ is a point or an interval.
I solved part (a) easily by the linearity of expectations but I can't solve part (b). I actually find it counter intuitive that the arbitrage-free price of a contingent claim can be an interval. Can someone please give a hint on part (b)? Please just a hint!
Some hints to (b):
In an incomplete market $\Pi(C)$ is an interval because we put lower and upper bounds on the arbitrage free prices. For example the upper bound on the price of a non-replicable $C$ is $$ \inf(E_{\mathbb{Q}}[X/(1+R)]:X\geq C, \: X \text{ is replicable}) $$ because if it would trade at a higher price than this, you could do arbitrage by selling $C$ and buying the portfolio replicating $X$.