Take $n\geq 1$, and $m_{ij}\in [0,1], 1\leq i,j \leq n$. Under what conditions is it possible to find measurable subsets $X_1,...,X_n$ of, say, $[0,1]$, such that $leb(X_i\cap X_j)=m_{ij}$?
Some relations are necessary, like $m_{ii}\geq m_{ij}$, or the fact that the matrix $(m_{ij})$ must be semi-definite positive, but it does not appear to be sufficient.
The same question holds with $m_{ij}\in \mathbb{R}_+, X_i \subset \mathbb{R}$.
The discrete version of this problem, where the $X_i$ are subsets of $\{1,2,\ldots,n\}$, is extremely difficult. For example, it includes the questions of the existence of Hadamard matrices and the existence of finite projective planes, which both remain unsolved despite a huge effort. So one can't expect to have necessary and sufficient conditions that are routine to check. It could be that the continuous problem is easier than the discrete one, though it isn't obvious to me. Actually I think the case when the $\{m_{ij}\}$ are integers is the same as the discrete problem since then each of the sets can be written as the disjoint union of atoms of measure 1.
Here is a partial idea where the last step needs a bit more justification which doesn't work as well as I had hoped, but may yet have promise.
As noted, this is a question about adding real numbers, no specialized measure theory is involved although the language is convenient. We will had hoped to define $2^n$ values $t_I$, one for each $I \subseteq [n]=\lbrace1,2,\cdots,n\rbrace $ so that if they are all non-negative, then the specified values can be achieved, otherwise they can not.
Suppose that we have a set $U$ of size (or measure) $m_{\emptyset}=|U|$ along with $n$ subsets $X_i$ for $i \in [n]$ each with complement $\overline{X_i}.$ For each $I \subseteq [n],$ let $m_I=|\cap_{i \in I}X_i|$ while $t_I=|(\cap_{i \in I}X_i) \cap (\cap_{j\notin I}\overline{X_j})|.$ As in the problem we can abbreviate $m_i=m_{i,i}$ for $m_{\lbrace i \rbrace}=|X_i|$ and $m_{i,j}$ for $m_{\lbrace i,j\rbrace}=|X_i\cap X_j|.$
If we know either set of values we can uniquely find the others: $$m_I=\sum_{I \subseteq J \subseteq [n]}t_J\hspace{0.1in} \text{ while } t_I=\sum_{I \subseteq J \subseteq [n]}(-1)^{|J|-|I|}m_J$$
If the $m_I$ are given, then the $t_I$ will be determined over the reals, but we want non-negative values. In the given problem we have only $n+\binom{n}{2}$ of the $m$ values, perhaps along with $m_{\emptyset}=1.$
So we first define the rest of the $m_I$ by $m_I=\min_{i,j \in I}m_{i,j}$ then solve for the $t_I$ and check that they are non-negative. I think that these choices will make $t_{[n]}$ as large as possible and thought that they would give all the $t_I$ the best chance to be non-negative consistent with the given information.
BUT now I notice problems with the simple case of asking for the sides of a triangle: $|U|=3$ $|X_1|=|X_2|=|X_3|=2$ and $|X_1 \cap X_2|=|X_1 \cap X_3|=|X_2 \cap X_3|=1$ If we do add the final condition $m_{\lbrace 1,2,3 \rbrace}=|X_1 \cap X_2 \cap X_3|=0$ then we do get the desired solution $t_{\{1,2\}}=t_{\{1,3\}}=t_{\{2,3\}}=1$ with the rest of the $t_I=0.$ HOWEVER if we make my suggested choice of $m_{\lbrace 1,2,3 \rbrace}=|X_1 \cap X_2 \cap X_3|=1$ then we do get $t_{\{1,2,3\}}=1$ as large as possible but then $t_{\{1,2\}}=t_{\{1,3\}}=t_{\{2,3\}}=0$ making $t_{\{1\}}=t_{\{2\}}=t_{\{3\}}=1$ and finally $t_{}=-1$
Think of the values $t_I$ as weights to be assigned to the regions of an ideal Venn diagram for $n$ sets. The conditions on the $m_i$ and $m_i,j$ (along perhaps with $m_{\emptyset}=1$) give $\frac{n^2+n}2$ (or else $\frac{n^2+n}2+1$ ) equations in $2^n$ variables $t_I$ along with the side condition that all the $t_I \ge 0.$ This is a linear programming problem. Perhaps there is a way to start with my assignmet and then adjust the values until success or proved failure, but I am not sure.
later The convex hull might be unworkable once the number of dimensions is $43$ or $43+\binom{43}{2}$ or $2^{43}$, but maybe not. I found the comments on integral problems pretty convincing and asked this question. However I subsequently answered it and realized that, while there is no projective plane of order $6$, We can achieve $m_{\emptyset}=43, m_i=7$ for $1 \le i \le 43$ and all $m_{i,j}=1$ by setting $|t_I|=0$ except that $|t_J|=1/\binom{41}{5}$ for all $\binom{43}{7}$ choices of $J$ with $|J|=7.$ So now I am back to thinking that there might be a simple algorithm which achieves the constraints with least $L_1$ error (so $0$ if possible.)
The set of symmetric $n\times n$ matrices $Q$ that one produces this way, is the convex polytope whose vertices are exactly the $2^n$ binary matrices $\chi_{A\times A}$, for $A\subset[n]:=\{1,\dots,n\}\, .$ I guess this object is well-known.
Given a family of $n$ subsets $X_i$ of $X:=[0,1]$, we may consider the usual construction of the refinement of it, defined by the $2^n$ disjoint sets $$\Xi_A:=\cap_{i\in A} X _ i\setminus \cup _ {i\notin A} X _ i\, , \qquad A\subset [n]\, .$$ Clearly, the $2^n$ numbers $m _ A:=\operatorname{meas}(\Xi_A)$ are non-negative real numbers summing to $1$; conversely, any point $\xi$ of the standard simplex $S$ spanned by the canonical basis $\{\operatorname{e} _ A\}_{A\subset [n]}$ of $ \mathbb{R}^{2^n}$ is produced this way by some family $\{X_i\}_{i\in[n]}$, since one can pass through a convenient family of disjoint sets $\Xi_A$ with $ \operatorname{meas}(\Xi_A)=\xi _ A$ and define $X_i:=\cup_{A\ni i} \Xi _ A $ .
Now consider the linear map $L:\mathbb{R}^{2^n}\ni \xi\mapsto m\in\mathbb{R}^{n^2}$ such that $m_{ij}:=\sum_{A\supset\{i,j\}} \xi _ A$. By the preceding observation, $L(S)=Q$, which proves that $Q$ is a convex polytope spanned by the $L(\operatorname{e} _ A)=\chi_{A\times A} $ for $A\subset[n]:=\{1,\dots,n\}\, .$
It remains to show that the $\chi_{A\times A}$ are convexly independent. Assume $\chi_{B\times B}$ is a convex combination of the $\chi_{A\times A}$ ,
$$ \chi_{B\times B}=\sum _ A c _ A \, \chi_{A\times A} \, .\qquad (1)$$ By restriction to the diagonal of $[n]\times[n]$ it follows $$ \chi _ B=\sum _ A c _ A \, \chi _ A \, ,\qquad(2)$$ so in particular $A\subset B$ for all $A$ corresponding to non-zero coefficients $c _A$ . Also, summing over all $i\in [n]$ in (1) $$|B|\\ \chi _ B = \sum _ A c _ A \, |A|\, \chi _ A , $$ and using (2) $$ \sum _ A c _ A \, \left (|B|- |A|\right) \, \chi _ A = 0, $$ and we conclude that the only non-zero coefficient is $c _ B =1$.
Rmk. Any atom-less probability space of course gives the same conclusion.
