We describe a widely applicable design-based framework for drawing causal inference in randomized experiments. Causal effects are defined as linear functionals evaluated at unit-level potential outcome functions. Assumptions about the potential outcome functions are encoded as function spaces. This makes the framework expressive, allowing experimenters to formulate and investigate a wide range of causal questions that previously could not be investigated with design-based methods. The framework is particularly suited for complex, non-discrete interventions and causal interference. We describe a class of estimators for estimands defined using the framework and investigate their properties. We provide necessary and sufficient conditions for unbiasedness and consistency. We also describe a class of conservative variance estimators, which facilitate the construction of confidence intervals. In order to demonstrate the value of our approach in practice, we provide several illustrative examples of causal investigations which can be handled within our framework, but that could not be addressed using conventional design-based methods.

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