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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

user90803
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2 Answers2

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

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

Community
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Batman
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    but the conditional expectation is defined for semi integrable functions, that is because Radon Nikodym holds for non-finite measures and that is the only thing you need for the construction of conditional expectations. – user90803 Jun 16 '14 at 06:13