Why Is It Acceptable to Interchange Sum and Integral for Martingale Proof?

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The discussion centers on the interchangeability of sums and integrals in the context of proving that M_t is a martingale, where M_t is defined using a series involving iid Poisson processes. The key point raised is the need to justify the interchange of the sum and integral, specifically using the dominated convergence theorem. This theorem allows for such exchanges under certain conditions, particularly when the series converges in the L^2 sense. Participants emphasize the importance of ensuring that the conditions for applying the theorem are satisfied. The conversation ultimately seeks clarity on the mathematical justification for this interchange in the context of martingale theory.
osprey
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Let M_t = \sum_{i=1}^\infty \frac{1}{i} (N_t^i - \lambda t) where \{ N^i \} is a sequence of iid Poisson processes with intensity \lambda. It can be shown that the series converges in the L^2 sense. Why is it ok to write

\int \left ( \sum_{i=1}^\infty \frac{1}{i} (N_t^i - \lambda t) \right ) dP = \sum_{i=1}^\infty \frac{1}{i} \int (N_t^i - \lambda t) dP?

(I need to show that M is a martingale and to do so, I would like to interchange the sum and the integral, but I cannot find an argument that seems to work...)

Thank you in advance for your help! :-)

/O
 
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