The notion of causal effect is fundamental across many scientific disciplines. Traditionally, quantitative researchers have studied causal effects at the level of variables; for example, how a certain drug dose (W) causally affects a patient's blood pressure (Y). However, in many modern data domains, the raw variables-such as pixels in an image or tokens in a language model-do not have the semantic structure needed to formulate meaningful causal questions. In this paper, we offer a more fine-grained perspective by studying causal effects at the level of events, drawing inspiration from probability theory, where core notions such as independence are first given for events and sigma-algebras, before random variables enter the picture. Within the measure-theoretic framework of causal spaces, a recently introduced axiomatisation of causality, we first introduce several binary definitions that determine whether a causal effect is present, as well as proving some properties of them linking causal effect to (in)dependence under an intervention measure. Further, we provide quantifying measures that capture the strength and nature of causal effects on events, and show that we can recover the common measures of treatment effect as special cases.

翻译：因果效应的概念是众多科学学科的基础。传统上，定量研究者通常在变量层面研究因果效应；例如，某种药物剂量（W）如何因果性地影响患者的血压（Y）。然而，在许多现代数据领域中，原始变量——如图像中的像素或语言模型中的标记——缺乏提出有意义因果问题所需的语义结构。本文受概率论启发，从事件层面研究因果效应，提供了一种更细粒度的视角；在概率论中，独立性等核心概念首先在事件和σ-代数中定义，随后才引入随机变量。在因果空间的测度理论框架内——这是最近提出的因果性公理化体系——我们首先引入若干二元定义以判定因果效应是否存在，并证明这些定义在干预测度下将因果效应与（不）独立性关联的性质。进一步，我们提出了量化测度以捕捉事件上因果效应的强度与性质，并证明常见的处理效应测度可作为其特例予以恢复。