Estimating integrals and measures over Hilbert space using finite dimensional projections

Question

Let $H$ be a separable Hilbert space with orthonormal basis $\{e_k\}_{k \in \mathbb{N}}$. Let $P_n$ be the projection onto the $n$-dimensional subspace of $H$: $$P_n x = \sum_{i=1}^n \langle x, e_i\rangle e_i, \quad x \in H.$$

Is it possible to compute measures and integrals over $H$ using a limit of the operators $P_n$ as $n \rightarrow \infty$? That is, if $\mu$ is a probability measure over $H$ and $A$ is a measurable set in $H$, then could we estimate $\mu(A)$ using the $P_n$? If so, how is this done?

Similarly, suppose we have either the Bochner integral $$\int_H x d\mu(x) \tag{1}$$ or equivalently the scalar-valued integral $$\int_H \langle x, h\rangle d\mu(x)$$ is it possible to compute these integrals as measures over finite dimensional subspaces of $H$ using the projections? For example, I have in mind something like: $$\int_H \lim_{n \rightarrow} P_n x d\mu(x)$$ but there are a number of issues with this approach. First, to compute (1) using the above integral one would have to interchange the integral and the limit but this is nontrivial. Second, how do these projections affect the infinite dimensional measure $d\mu$? Do they become something like a measure over $P_nH$, and if so, how are these defined?

Answer 1

Interchanging the limit and the integral is fine in this case. Indeed, using the definition of Bochner integrals, by approximating with simple functions, one easily observes that Bochner integrals can be interchanged with the application of bounded linear operators, so in particular,

$$\int P_nx \, d\mu(x) = P_n\int x \, d\mu(x) \to \int x \, d\mu(x)$$

in norm.

This, of course, assumes $x$ is integrable w.r.t. $\mu$ - since we’re in a separable space, this is just the same as saying the Lebesgue integral $\int \|x\| \, d\mu(x) < \infty$. Note that this automatically implies $P_nx$ is integrable for all $n$. Also note that when $x$ is not integrable, the integral $\int x \, d\mu(x)$ does not make sense, and $P_nx$ could be non-integrable as well - $\mu$ could be supported on $P_nH$ for some $n$ and still $x$ could be non-integrable. (Even in dimension $1$, not all probability distributions have a well-defined finite mean.)

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Projections don’t send measurable sets to measurable sets in general, so $P_nA$ isn’t necessarily measurable, and thus it’s not possible to consider $\mu(P_nA)$. A non-null set $A$ could also be disjoint from $P_nH$ for all $n$, so considering $A \cap P_nH$ won’t help in approximating $\mu(A)$. As such, there seems to be no way of approximating $\mu(A)$ using the projections $P_n$.

Though, inducing measures on $P_nH$ is easy: $P_n$ is a continuous and thus Borel map from $H$ to $P_nH$, so the pushforward probability measure $P_n^\ast \mu$ makes sense.