Data spaces are emerging as a transformative approach to data management, sharing, and governance. A data space ecosystem typically has entities that produce, process, and consume data as well as contract, consent, and identity management services. The aspect of data quality, or, more generally, data veracity, is often overlooked. However, it is becoming increasingly clear that for healthy, functioning data spaces, ensuring the veracity of the data shared will be an essential service. In this paper, we introduce the novel notions of data veracity and data veracity assurance: a holistic notion of data quality in data spaces and a conceptual framework for its end-to-end assurance in data space service chains. Our approach is compatible with the leading data space frameworks, supports personal data sharing, and is being implemented as a set of open source building blocks. We define two major complementary regimens of data veracity assurance: presentations of trust-based Attestations of Veracity and the trustless checking of formal Proofs of Veracity. We employ verifiable credentials to bridge the gap between technical notions of data quality and legally binding contracts in a systematic way. Verifiable credentials also enable the application of our proposed framework in radically different trust settings, from closed, intermediaryorchestrated data spaces to fully decentralized ones and cross-data-space data exchange scenarios.