TITLE: How to interpret 2 variables separated by a comma in chained inequalities
QUESTION [15 upvotes]: What does $0\le x,y\le1$ mean? See the red circled part in the image below for an example.

I first thought it meant: $x\ge0$ and $y\le1$
Then I thought it meant: $0\le x\le y \le 1 $
But, based on the green part, I believe it means: $x$ and $y$ are in $[0,1]$

Is this notation unambiguous? In probability, the comma means $\bigcap$, so, to me, these are "separate" statements, as in (1) and not (3).

REPLY [14 votes]: It is a convention for $0\leq x \leq 1$ and $0 \leq y \leq 1$ and is mostly to avoid typing it out twice. The comma is used this way in the equivalent statement $x,y \in [0,1]$ as well so it's consistent with that notation.

REPLY [4 votes]: The overall integral of $f_{X,Y}(x,y)$ has to be one. Let's then compare the different assumptions.
(a)
The integral
$$\int_0^{\infty}\int_{-\infty}^1x+\frac32y^2\ dy\ dx$$
is not convergent.
(b)
$$\int_0^{1}\int_{x}^1x+\frac32y^2\ dy\ dx=\int_0^1x+\frac12-x^2-\frac12x^3\ dx=\frac{13}{24}\not=1.$$
(c)
$$\color{green}{\int_0^{1}\int_{0}^1x+\frac32y^2\ dy\ dx=\int_0^1x+\frac12\ dx=1}.$$
So, interpretation (c) seems to be correct.