# Uniqueness of 1-dimensional distributions via uniqueness of Laplace Transform

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.