  We tackle the extension to the vector-valued case of consistency results for
Stepwise Uncertainty Reduction sequential experimental design strategies
established in [Bect et al., A supermartingale approach to Gaussian process
based sequential design of experiments, Bernoulli 25, 2019]. This lead us in
the first place to clarify, assuming a compact index set, how the connection
between continuous Gaussian processes and Gaussian measures on the Banach space
of continuous functions carries over to vector-valued settings. From there, a
number of concepts and properties from the aforementioned paper can be readily
extended. However, vector-valued settings do complicate things for some
results, mainly due to the lack of continuity for the pseudo-inverse mapping
that affects the conditional mean and covariance function given finitely many
pointwise observations. We apply obtained results to the Integrated Bernoulli
Variance and the Expected Measure Variance uncertainty functionals employed in
[Fossum et al., Learning excursion sets of vector-valued Gaussian random fields
for autonomous ocean sampling, The Annals of Applied Statistics 15, 2021] for
the estimation for excursion sets of vector-valued functions.
