I know that the Kaplan-Meier estimator is biased because my textbook says so. However, I don't understand why the following proof doesn't work:
Let $\hat{S}(t)$ be the Kaplan-Meier estimate for the survival function $S(t)\equiv P(T_i > t)$ where $T_i$ are iid failure times. Let $\hat{\Lambda}(u)$ be the Nelson-Aalen estimator for the cumulative hazard function $\Lambda(u)$.
It is known that $\frac{\hat{S}(t)}{S(t)}-1 = -\int\limits_{0}^{t}\frac{\hat{S}(u^-)}{S(u)}d\{\hat{\Lambda}(u)-\Lambda(u)\}$.
Now, $\int\limits_{0}^{t}\frac{\hat{S}(u^-)}{S(u)}d\{\hat{\Lambda}(u)-\Lambda(u)\}$ is a martingale because $\hat{\Lambda}(u)-\Lambda(u)$ is a martingale and because $\frac{\hat{S}(u^-)}{S(u)}$ is a predictable process.
So, $\mathbb{E}[\frac{\hat{S}(t)}{S(t)}-1]=0 \Rightarrow \mathbb{E}[\frac{\hat{S}(t)}{S(t)}]=1 \Rightarrow \mathbb{E}[\hat{S}(t)]={S(t)}$ since $S(t)$ is a non-stochastic function.