I have been reading the paper ['Improved Bounds for the Sunflower Lemma' (Ann. of Math., Vol. 194(3), pp. 795-815)](https://annals.math.princeton.edu/2021/194-3/p05), and have not managed to understand the following:


1. I would like a formalization for what a $p$-biased distribution (pg. 796) is, since $R$ is treated both as a set and a random-variable in Definition 1.5 (pg. 796) and I am not certain how this is done.  

What I have tried: Consider the measure space $(X, \mathcal{P}(X), \mathbb{P})$ and the measurable space $(X, \mathcal{P}(X))$. We define the random-variable $\mathcal{R}: X \to X$ such that $\mathcal{R}: t \mapsto t$, and the probability-measure $\mathbb{P}: \mathcal{P}(X) \to [0, \infty]$, such that $\forall \, t \in X: \mathbb{P}(\{t\}) = p$, and $\forall \, S \subseteq X:$ $\mathbb{P}(S) := \prod\_{s \in S} \mathbb{P}(\{s\}).$ Then, the distribution function is as follows- $\forall \, A \in \mathcal{P}(X):$ $$\mathbb{P}^{X}(A) = \mathbb{P}(\{w \in X: \mathcal{R}(w) \in A\}) = \mathbb{P}(A).$$ Is this indeed the formal definition of a $p$-biased distribution? If so, what does Definition 1.5 mean (, since $R$ is considered both as a r.v. and a set in the definition)? If not, what is a $p$-biased distribution formally?
2. Assuming a heuristic understanding of $U(X,p)$ as being as stated by the authors on pg. 796 before Definition 1.5, we consider Lemma 1.6. On pg. 797, the paper claims that by 'the union bound', we have that $$\textrm{Pr} \: [\forall \, i \in [r], \, \exists \, S \in \mathcal{F}: S \subseteq Y\_{i}] > 0$$, where $Y\_{i}$ is the set of all points in $X$ coloured with colour $i$. I am not able to follow this, since I am not able to say anything non-trivial about $\textrm{Pr} (\cup\_{i \in [r]} \varepsilon\_{i})$.
3. When the authors define the weight function $\sigma$ on pg. 799, they restrict its domain to $\mathcal{F}$, but on pg. 800, they refer to $\sigma(\mathcal{F}\_T)$, which doesn't make sense to me, since it doesn't appear that $\forall \, T \subseteq X, \, T \neq \emptyset: \mathcal{F}\_T \subset \mathcal{F}.$ Therefore, I was wondering if we require that the domain of $\sigma$ be $\mathcal{P}(X)$ instead.
4. On pg. 800, after Definition 2.1, the authors state that if $(\mathcal{F},\sigma)$ is s-spread, where $s= (s\_{0}; s\_{1}, \dots, s\_{w})$ is a weight-profile, in particular, $\mathcal{F}$ is a $w$-set-system.
What I am able to show is that if indeed $\mathcal{F}$ contains a set $S$ with $|S| > w$, then, we have $$\forall R \in \mathcal{F}, S \subset R: \sigma (R \, \backslash \, S) = 0.$$ What I thought I could try showing was that $\sigma$ would be $0$ everywhere, which is not allowed, by definition of a weight function on pg. 799.


Any help on any or all of these questions is appreciated.



On 'Improved Bounds for the Sunflower Lemma' [Alweiss, Lovett, Wu, Zhang]
1. The random variable $R$ here takes values in the power set ${\mathcal P}(X)$ of $X$, not in $X$: it's a random *set* in $X$, not a random *point*. For the purposes of this argument, the only important features of $R$ are that the events $(x \in R)$ for each $x \in X$ are independent events of probability $p$. But if you really wanted to formally model this random variable by a measurable map on a probability space as you attempted to do above, one could do so by taking the probability space $({\mathcal P}(X), {\mathcal P}({\mathcal P}(X)), {\mathbb P})$ with the probablity measure
$$ {\mathbb P}(\{R\_0\}) = p^{\# R\_0} (1-p)^{\# (X \backslash R\_0)}$$
for all $R\_0 \subset X$ (i.e., $R\_0 \in {\mathcal P}(X))$), and take $R: {\mathcal P}(X) \to {\mathcal P}(X)$ to be the identity map $R: R\_0 \mapsto R\_0$. However, the precise probability space model for this random set $R$ is not of particular relevance; see [this post of mine](https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/) for some further discussion.
2. Once one can prove that ${\mathbb P}(E\_i) > 1-1/r$ for all $i=1,\dots,r$, the union bound (applied to the complements of $E\_i$) shows that ${\mathbb P}(\bigwedge\_{i=1}^r E\_i) > 0$, thus the $E\_i$ simultaneously hold with positive probability.
3. Presumably this will become clearer at the later stages of the argument where the quantity $\sigma({\mathcal F}\_T)$ is actually used, but I would guess that here the authors are implicitly abusing notation by identifying the link ${\mathcal F}\_T = \{ S \backslash T: S \in {\mathcal F}, T \subset S \}$ with the associated subset $\{ S: S \in {\mathcal F}, T \subset S \}$ of ${\mathcal F}$ for the purpose of computing weights.
4. Apply Definition 2.1(ii) with $T$ replaced by your set $S$ with $|S| > w$ and with the conventions indicated in my response to item 3.


More generally, you might like to look at my [general advice on how to read mathematical papers](https://terrytao.wordpress.com/advice-on-writing-papers/on-compilation-errors-in-mathematical-reading-and-how-to-resolve-them/) (in particular how to resolve issues like 3, 4 that are often coming from some typo or implicit convention on the author's part).


