Assume that values (e.g., probabilities or cardinals) of a measure on a finite set $\Omega$ are given for sets $A_1,\dots, A_n$ and all of their intersections $A_i, A_i\cap A_j, A_i\cap A_j\cap A_k, ...$. My question:

Is there any necessary and sufficient condition that can be simply checked to make sure that there is a measure on the set $\Omega$ that can produce the given numbers.

I came across this problem after solving this counting question, where I realized the numbers given in the source textbook cannot be correct, and showed it by checking two sufficient conditions.

I am wondering if there is any efficient algorithm for this purpose. I guess the above problem should be NP-hard since I can formulate it as a linear programming model for the case of real-valued measures (such as probability measures), which includes exponential number of linear constraints. I think the problem is more difficult for integer-valued measures.

Assume that values (e.g., probabilities or cardinals) of a measure on a finite set $\Omega$ are given for sets $A_1,\dots, A_n$ and all of their intersections $A_i, A_i\cap A_j, A_i\cap A_j\cap A_k, ...$. My question:

Is there any necessary and sufficient condition that can be simply checked to make sure that there is a measure on the set $\Omega$ that can produce the given numbers.

I came across this problem after solving this counting question, where I realized the numbers given in the source textbook cannot be correct, and showed it by checking two sufficient conditions.

I am wondering if there is any efficient algorithm for this purpose. I guess the above problem should be NP-hard since I can formulate it as a linear programming model for the case of real-valued measures (such as probability measures), which includes exponential number $O(2^n)$ of linear constraints involving $|\Omega|$ variables. I think the problem is more difficult for integer-valued measures.

Assume that values (e.g., probabilities or cardinals) of a measure on a finite set $\Omega$ are given for sets $A_1,\dots, A_n$ and all of their intersections $A_i, A_i\cap A_j, A_i\cap A_j\cap A_k, ...$. My question:

Is there any necessary and sufficient condition that can be simply checked to make sure that there is a measure on the set $\Omega$ that can produce the given numbers.

I came across this problem after solving this counting question, where I realized the numbers given in the source textbook cannot be correct, and showed it by checking two sufficient conditions.

I am wondering if there is any efficient algorithm for this purpose. I guess the above problem should be NP-hard since I can formulate it as a linear programming model for the case of real-valued measures (such as probability measures), which includes exponential number of constraints. I think the problem is more difficult for integer-valued measures.

Assume that values (e.g., probabilities or cardinals) of a measure on a finite set $\Omega$ are given for sets $A_1,\dots, A_n$ and all of their intersections $A_i, A_i\cap A_j, A_i\cap A_j\cap A_k, ...$. My question:

Is there any necessary and sufficient condition that can be simply checked to make sure that there is a measure on the set $\Omega$ that can produce the given numbers.

I came across this problem after solving this counting question, where I realized the numbers given in the source textbook cannot be correct, and showed it by checking two sufficient conditions.

I am wondering if there is any efficient algorithm for this purpose. I guess the above problem should be NP-hard since I can formulate it as a linear programming model for the case of real-valued measures (such as probability measures), which includes exponential number of linear constraints. I think the problem is more difficult for integer-valued measures.