Suppose that $X$ is a Banach space. Denote $\gamma$ as a nondegenerate Gaussian probability measure on $X$ with mean $0$.
Question: Is it true that
$$\int_X{d\gamma(t)}=0?$$
Or we have
$$\int_X{d\gamma(t)}=1?$$
2026-04-13 06:37:07.1776062227
Integration w.r.t nondegenerate Gaussian probability measure on $X$ with mean $0$
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Since $\gamma$ is a probability measure, you have $$\gamma(X) = \int_X \mathrm{d}\gamma(t) = 1$$ and the mean is $$\int_X t \, \mathrm{d}\gamma(t) = 0.$$ (This is the zero element of $X$).