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.

翻译：我们解决了[Bect等，基于高斯过程的序贯实验设计的超鞅方法，Bernoulli 25, 2019]中建立的逐步不确定性缩减序贯实验设计策略一致性结果向向量值情形的推广。这首先促使我们阐明：在紧指标集假设下，连续高斯过程与连续函数Banach空间上高斯测度之间的联系如何推广至向量值设定。由此，上述论文中的若干概念和性质可被直接扩展。然而，某些结果因向量值设定而复杂化，这主要源于伪逆映射缺乏连续性，影响了基于有限点观测的条件均值与协方差函数。我们将所得结果应用于[Fossum等，面向自主海洋采样的向量值高斯随机场极值集学习，应用统计年鉴15, 2021]中用于估计向量值函数极值集的集成伯努利方差与期望测度方差不确定性函数。