Using my professor's notation, Ito integral is defined as
$$ \int_{t_0}^tG_sdW_s=l.i.m._{h\to0}\sum G_{t_1}(W_{t_{i+1}}-W_{t_i}) $$
where $G_s$ is a general stochastic process and $W_s$ is a Wiener process. Using this definition, he proved that expectation of Ito integral is 0 by
$$ \begin{align*} E\left\{\int_{t_0}^tG_sdW_s\right\}&=E\left\{\lim_{h\to0}\sum G_{t_i}(W_{t_{i+1}}-W_{t_i})\right\}\\ &=\lim_{h\to0}\sum E\{G_{t_i}(W_{t_{i+1}}-W_{t_i})\}\\ &=\lim_{h\to0}\sum E\{G_{t_i}\}E\{W_{t_{i+1}}-W_{t_i}\}=0 \end{align*} $$
Similarly, Stratonovich integral was defined as
$$ \int_{t_0}^tG_sdW_s=l.i.m._{h\to0}\sum G_{\frac{t_{i+1}+t_i}{2}}(W_{t_{i+1}}-W_{t_i}) $$
And he left a note saying that relation $E\left\{\int_{t_0}^tG_sdW_s\right\}=0$ does not hold for Stratonovich integral. However, if I expand left-hand side similar to Ito integral but using Stratonovich integral definition I get
$$ E\left\{\int_{t_0}^tG_sdW_s\right\}=\lim_{h\to0}\sum E\left\{G_{\frac{t_{i+1}+t_i}{2}}\right\}E\{W_{t_{i+1}}-W_{t_i}\} $$
Why does relation $E\left\{\int_{t_0}^tG_sdW_s\right\}=0$ does not hold for Stratonovich integral?
Note that $G_{\frac{t_{i+1}+t_i}{2}} := \frac{G_{t_{i+1}}+G_{t_i}}{2}$, such that $$ \mathbb{E}\left[\int_{t_0}^tG_S\,\mathrm{d}W_s\right] = \lim_{h\rightarrow0} \frac{1}{2} \sum_i \mathbb{E}[(G_{t_{i+1}}+G_{t_i})(W_{t_{i+1}}-W_{t_i})]. $$ Now, let's recall that the increments of a Wiener process are independent of past values, hence $\mathbb{E}[G_{t_i}(W_{t_{i+1}}-W_{t_i})] = 0$ but $\mathbb{E}[G_{t_{i+1}}(W_{t_{i+1}}-W_{t_i})] \neq 0$, because $W_{t_{i+1}}-W_{t_i}$ is the next increment with respect to $G_{t_i}$ but the previous one with respect to $G_{t_{i+1}}$.