\int_F\exp(\xi X)dP&=&\sum_{j=1}^n\int_{F\cap A_j}\exp(a_jX)dP+P(F\cap N)\\ &=& \sum_{j=1}^n \exp\bigg( \frac{1}{2}a_j^2t\bigg)P(A_j\cap F)+P(F\cap N)
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