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Consider a stochastic process $X = (X_t)_{t \geq 0 }$ on $\mathbb{R}$ and two probability measures $\mathbb{P}_1,\mathbb{P}_2$ with expectations denoted $\mathbb{E}_1,\mathbb{E}_2$. Suppose for $g$ continuous we have the following

$$ \mathbb{E}_1[\int_0^\infty \exp(-\lambda t)g(X_t)\,dt] = \mathbb{E}_2[\int_0^\infty \exp(-\lambda t)g(X_t)\,dt]. $$

How can we show using this result that the one-dimensional distributions of $X$ under $\mathbb{P}_1,\mathbb{P}_2$ are the same? I know it has something to do with uniqueness of the Laplace transform, but I am unsure how to apply this.

Thanks.

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