Data distribution definition and notation, what is wrong or right?

My work area is machine learning but since I am not from a math background I am struggling a lot to do the right job.

I wanna define the data distribution which is defined for data points. I have rewritten the following definition many times but my professor is still unhappy with my definition. I am confused if this is the correct definition:

Definition (Distribution): We consider a collection of $$T$$ data distributions $$\mathcal{D}=\{\mathcal{D}_1,\mathcal{D}_2,\dots,\mathcal{D}_T\}$$ as the set of $$T$$ mappings each indexed by $$t \in \mathbb{N}$$ and denoted as $$\mathcal{D}_t(\cdot):\mathbb{R}^{d_t}\times\mathbb{R}^{c_t}\rightarrow\mathbb{R}^+\cup\{0\}$$ s.t. $$\int_{\mathbb{R}^{d_t}\times\mathbb{R}^{c_t}}\mathcal{D}_t(\cdot)dx^tdy^t=1$$ where $$(x^t,y^t)\sim\mathcal{D}_t$$.

We consider that $$t$$th distribution is denoted as $$\mathcal{D}_t$$ from which dataset samples $$(x^t_i,y^t_i)$$'s are drawn, where $$x^t_i \in \mathbb{R}^{d_t}$$, $$y^t_i \in \mathbb{R}^{c_t}$$, and $$d_t,c_t\in \mathbb{N}$$.

I have three questions:

1. Is my definition correct? if not, what is the right definition?
2. Can we have distribution from one data point $$(x^t_i,y^t_i)\sim\mathcal{D}_t$$? how to write the density function for that?
3. Is there any book or resources that I can refer to for my math definitions and notations to learn more about the rules for mathematical notations?
• To me each $\mathcal{D}_t$ is a correctly defined probability density on $\mathbb{R}^{d_t}\times\mathbb{R}^{c_t}$ . For one data point the meaning of $\mathcal{D}_t(x_i^t,y_i)dx_i^tdy_i^t$ is the probability that a random sample lies in a small interval around that point (I would probably omit the sub and superscripts here): $\mathcal{D}_t(x,y)dxdy$ is the same. Any introductory book to probability theory should cover these basic concepts. Sep 21, 2022 at 11:58