Consider the following problem:
Suppose we know that $f_X(x)=1$, now I want to calculate the autocorrelation:
$E[\underbrace{X(t)X(t+\tau)}_{g(t)}]=\int_0^1 g(t)f_X(x)dx=\int_0^1 X(t)X(t+\tau))dx=\color{red}{\int_0^1g(t-\lambda)g(t+\tau-\lambda)d\lambda}$
My question is the red part; since the random variable is $T$ and in the integral I let $\lambda$ to be a dumb variable representing $T$, so I get the red part.
I know the red part is correct according to the answer; however, how could I do this by the rules of integral, i.e. whether the last equality is true?