Defining a measure on the space of infinite words over a finite alphabet All words in this question are over the alphabet $\{0,1\}$.
Let $X$ be the set of words of infinite length.
For a finite word $w$, write $w0$ for the word obtained by appending $0$ to the end of $w$. Define $w1$ similary.
For each finite word $w$, fix a nonnegative real number $m_w$. Assume that $m_w=m_{w0}+m_{w1}$ for each word $w$. Assume that $m_\epsilon=1$, where $\epsilon$ is the unique word of length zero.
Endow $\{0,1\}$ with the discrete topology and $\{0,1\}^\mathbb{N}$ with the product topology. The latter space is naturally identified with $X$, making $X$ into a topological space.
For a finite word $w$, write $A_w\subset X$ for the set of infinite words that start with $w$.
Is there a countably-additive Borel measure $\mu$ on $X$ such that $\mu(A_w)=m_w$ for every finite word $w$? Is it unique?
It seems to me that the answer must be positive, but I'm still struggling with measure theory.

Yes, there are probability measures on sequence space satisfying your conditions.  For instance, if $m_w=2^{-|w|}$, where $|w|$ is the length of $w$. And many many others.  Part of the key is the Kolmogorov extension theorem, which in your case results in the $m_w$ for all finite length words inducing a measure on sequence space.  More generally, any $0,1$-valued discrete time  stochastic process gives an example.
