Why martingales need to be integrable?

Question

In almost every book the definition of martingale ask for integrability of every random variable. Why this property is needed? If we remove it the property that the expected values is constant holds?

Thanks! any help will be appreciated

Answer 1

A random variable that is not integrable has no expected value.

Answer 2

A martingale is defined in terms of conditional expectations - $\{X_n\}_{n\in \mathbb{N}}$ is a martingale wrt the filtration $\{\mathcal{F}_n\}_{n \in \mathbb{N}}$if the following conditions hold:

i) $X_n$ is adapted to $\mathcal{F}_n$

ii) $E|X_n|<\infty$ (this is necessary for (iii) to be defined, by the definition of conditional expectation)

iii) $E[X_{n+1} \mid \mathcal{F_n}] = X_n$ ${{{{}}}}$