# Math Help - Markov Chain Conditional Expectation

1. ## Markov Chain Conditional Expectation

Hello, in relation to Markov chains, could you please clarify the following equations:

In particular, could you please expand on why the first line is equal. Surely from $\100dpi \mathbb{P}(A)=\mathbb{E}[\mathbf{1}_A(\omega )]$, along with the first equation, this implies that:

I just don't see why they are all equal. Please could you provide some intuition on this. Thanks

2. Originally Posted by tunaaa
Hello, in relation to Markov chains, could you please clarify the following equations:

In particular, could you please expand on why the first line is equal. Surely from $\100dpi \mathbb{P}(A)=\mathbb{E}[\mathbf{1}_A(\omega )]$, along with the first equation, this implies that:

I just don't see why they are all equal. Please could you provide some intuition on this. Thanks
The first two are equal by definition. The actual meaning of conditioning is introducing a new measure i.e. $\mathbb{P}^1(\cdot)=\mathbb{P}(\cdot | A)$. When you take expectation with respect to that, you indicate it by $\mathbb{E}[\cdot|A]$.

In your calculation your indicator function is meaningless. What is the set operation $A | B$?