It can be easily shown that for the process $X=(X_t)_{t\geq 0}$ which has independent and stationary increments the following holds: $$ EX_n = nEX_1, $$ where $n$ is an integer number.
Is this equation true if we replace an integer $n$ by a real number $t$, i.e. $$ EX_t = t EX_1, $$ If is it true, how we can prove it?
If $(X_t)$ has stationary independent increments the the function $f(t)=EX_t$ satisfies the Cauchy equation $f(t+s)=f(t)+f(s)$. If you assume that the process $(X_t)$ is a measurable process you can show that $f$ is measurable and this implies $f(t)=ct$ for all $t$ where $c$ is a constant. Hence $EX_t=ct$ so $EX_t=tEX_1$.
The equation $f(t+s)=f(t)+f(s)$ comes from $X_{t+s}= (X_{t+s}-X_s) +X_s$