Uniqueness of Brownian Motion from properties

Question

I am trying to understand how the following properties for a stochastic process

1. $B_t$ is a Gaussian process
2. $B_t$ has independent increments
3. $t\to B_t(\omega)$ is continuous for almost all $\omega$

Uniquely determine that a process is a Brownian process centered at some $x\in\mathbb{R}$. This post talks about knowing all the finite distributions will tell us that the measure for the stochastic process is uniquely generated by the set of all finite distributions: Uniqueness of Brownian motion. My guess is that I will need to show that these three properties uniquely determine the finite distributions, but I have no idea how to generate finite the finite distributions with these properties.

Edit: Does Kolmogorov's Extension Theorem imply that the measure generated is unique?