Relating Entropies of Probability Spaces

[Juanriq]

Ahoy hoy, trying to see if I am on the right track with a question.

Homework Statement

Let $T: (X, \mathcal{B}, m)$ to itself be an mpt of a probability space. Suppose $X = X_1 \cup X_2$ where $X_1 \cap X_1 = \emptyset$ and $m(X_i) > 0$ and also $T^{-1}X_i = X_i$. Let $T: (X_i, \mathcal{B}|X_i, m_i)$ to itself where $T_i = T|X_i$ and [latex[ m_i = m|X_i [/itex] normalized. What is the relation between $h_m(T)$ and $h_{m_i}}(T_i)$?

Homework Equations

The Attempt at a Solution

I know $m(X) = 1$, so say $m(X_1) = p$ and $m(X_2) = 1-p$. This would mean that $m_1 = \frac{m|X_1}{m(X_1)} = \frac{m|X_1}{p}$ and also $m_2 = \frac{m|X_2}{m(X_2)} = \frac{m|X_1}{1-p}$. Now, $h_m(T) = -m(X_1) \log m(X_2) - m(X_2) \log m(X_2)$ and so $h_{m_i}}(T_i) = -m_1(X_1) \log m_1(X_2) - m_2(X_2) \log m_2(X_2)$.



Is this right so far? Thanks!

 

[Nino MMP]

Yes, you are on the right track. The relation between h_m(T) and h_{m_i}(T_i) can be expressed as follows: h_m(T) = p h_{m_1}(T_1) + (1-p) h_{m_2}(T_2).