are given in Section \[section: Martingality of local martingales\]. Finally, in Section \[A Case Study - Martingality in Terms of SDEs driven by Hilbert-Space-Valued Brownian Motion\], we deduce explicit martingale conditions for stochastic exponentials driven by infinite-dimensional Brownian motion.

**A Short Remark Concerning Notations:** All non-explained notations can be found in the monograph of [@JS]. Moreover, we usually skip the terminology *up to indistinguishability*. *Equality of processes* as well as *uniqueness of processes* should be read up to indistinguishability.

A Simple Description of Martingality {#First important Observations}
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We start with a characterization of the martingality of non-negative local matingales, which is as elementary as useful. Fatou’s lemma implies that a non-negative local martingale is a supermartingale, which itself is a martingale if and only if it has constant expectation. Therefore, the martingality of non-negative martingales is a property that only depends on the expectation. The following two consequences take advantage of this important observation.

We fix a filtered probability space $(\Omega, \mathcal{F}, {\mathfrak{F}}, {{\mathbf{P}}})$, where ${\mathfrak{F}}= (\mathcal{F}_t)_{t \geq 0}$ is a right-continuous filtration, and a one-dimensional non-negative local $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale $Z$ with ${{\mathbf{P}}}$-a.s. $Z_0=1$. The assumption that $Z_0$ is ${{\mathbf{P}}}$-a.s. constant can be relaxed to the case where $Z_0$ 