Kolmogorovs Extension Theorem

Question

I'm dealing with Pardos Proof on page 13, which shows the construction of a selfsimilar process $X$ with independent increments, while there is only $\nu_{X(1)}$ given. I asked myself, how he does assure the following requirement for the application of Kolmogorovs Extension theorem: $$\nu_{t_1,\dots,t_{n+1}}\left(A_1,\dots,A_n,\mathbb{R}\right)=\nu_{t_1,\dots,t_{n}}\left(A_1,\dots,A_n\right), $$ where the only things whose existence we know are the 1-dimensional marginal distribution $\mu_t$ and that of the future increments $\mu_{s,t}$ and $$\mu_t=\mu_s * \mu_s,t\quad\&\quad\mu_{r,s} * \mu_{s,t}=\mu_{r,t}. $$ I tried to construct the measure $\nu_{t_1,\dots,t_{n+1}}$ and came up with the following (where ' stands for transposed) $$\nu_{t_1,\dots,t_n}=\ \nu_{\left(X_{t_1},\dots,X_{t_n}\right)'}=\nu_{\left(X_{t_1}-X_{t_0}\right)\left(1,\dots,1\right)'+\left(X_{t_2}-X_{t_1}\right)\left(0,1,\dots,1\right)'+\dots+\left(X_{t_n}-X_{t_{n-1}}\right)\left(0,\dots,0,1\right)'},$$ as well as $$ \nu_{\left(X_{t_k}-X_{t_{k-1}}\right)\left(0,\dots,0,1,\dots, 1\right)^T}\left(A_1\times\dots\times A_n\right)=\delta_0\left(\bigcap_{i=1}^{k-1} A_i\right)\mu_{t_{i-1},t_i}\left(\bigcap_{i=k}^{n} A_i\right).$$ From this I deduced $$ \nu_{t_1,\dots,t_n}\left(A_1\times\dots\times A_n\right)\\ =\ \int_{\mathbb{R}^n} \nu_{\left(X_{t_k}-X_{t_{k-1}}\right)\left(0,\dots,0,1,\dots, 1\right)^T}\left(\left(A_1\times\dots\times A_{n-1}\right)-y\right)\quad\nu_{X_{t_1},\dots,X_{t_{n-1}},X_{t_{n-1}}}\left(\text{d}y\right)\\ =\int_{\mathbb{R}^n} \prod_{i=1}^{n-1}\delta_0\left(A_i-y_i\right)\nu_{\left(X_{t_n}-X_{t_{n-1}}\right)\left(0,\dots,0,1\right)^T}\mu_{t_n-t_{n-1}}\left(\mathbb{R}-y_n\right)\quad\nu_{X_{t_1},\dots,X_{t_{n-1}},X_{t_{n-1}}}\left(\text{d}y\right)\\ =\int_{\mathbb{R}^n} 1_{\left\{A_1\times\dots\times A_{n-1}\right\}}\left(\left(y_1,\dots,y_{n-1}\right)^T\right)\quad\nu_{X_{t_1},\dots,X_{t_{n-1}},X_{t_{n-1}}}\left(\text{d}y\right) $$ by convolution. I have the feeling I'm stuck. Does anybody have an idea? I'd be really thankful if one of you could help me out!