I am following some course notes on Measure Theory and Integration that defer the reader to Folland as an accompanying textbook. One of the hand-written problems in the notes is introduced with
Let $f \in \mathscr{L}^{1}(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$, $\lambda$ the Lebesgue measure.....
I am familiar with the following definitions from Folland:
$$\mathcal{L}^{1}(\mu) = \left\{f: X \to \overline{\mathbb{R}}: f \text{ measurable }, \int |f|d\mu < \infty \right\}$$ $$ L^{1}(\mu) = \left\{[f]: f \in \mathcal{L}^{1}(\mu)\right\} = \mathcal{L}^{1}(\mu) / \mathcal{N} $$
But I am not sure what exactly $f \in \mathscr{L}^{1}(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$ ought to mean. I could make some educated guesses, but for the purpose of the problem I am inclined to think that $f$ needs to be non-negative, which isn't in either of the above definitions.
Is this a well known notation, and if so, what does it mean?
This notation is not universally standard and I can't say for sure what it means without more context, but it seems extremely likely to me that this is just a synonym for either $\mathcal{L}^1(\lambda)$ or $L^1(\lambda)$ (taken literally, if $f$ is supposed to be a function, then it would be in $\mathcal{L}^1$, but it is common to abuse notation and identify functions with their equivalence classes, and indeed many authors don't even have a notation for what Folland calls $\mathcal{L}^1$). Here $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ presumably is supposed to be the measure space of Lebesgue measure, $\mathcal{B}(\mathbb{R})$ being the Borel $\sigma$-algebra on $\mathbb{R}$. So this notation is just mentioning the entire measure space (underlying set, $\sigma$-algebra, and measure), rather than just the measure as in Folland's notation.