Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions.

However, I have the following statement from a set of notes:
$$g(t,S_t) = E[(S_T-K)^{+}\mid F_t]$$

Where $g$ is some Borel function.

How can this be justified? Our $f$ here is $(x-K)^+$ which is not bounded.

Thank you

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