Conditional Expectation vs Conditional Probability Markov Process

Question

For my stochastic process class, we have many definitions of a Markov Process. One of them is the following:

$P(X_{t+1} \in A | \sigma(X_1, ...,X_t)) = P(X_{t+1} \in A | \sigma(X_t))$.

(A measurable)

I have seen it stated that this is equivalent to

$E(f(X_{t+1}) | \sigma(X_1, ...,X_t)) = E(f(X_{t+1}) | \sigma(X_t))$.

for bounded, measurable f

I am having trouble seeing why this equivalence is true. Could someone shed some light on this?

Thanks!

Answer 1

The first equation follows from the second by taking $f=I_A$.

If the first equation holds then the second one holds for $f$ of the form $f=I_A$ with $A$ measurable. Hence it holds for all simple functions $f$. Any bounded measurable function is a uniform limit of simple functions. Also Bounded Convergence Theorem holds for conditional expectations. Hence the second equation holds for all bounded measurable functions $f$.