Classification with Fuzzy Class Labels

I am currently involved in a project involving fuzzy class labels.

To be clear, whereas classes are discrete and mutually exclusive in a typical binary classification task, the classes I am working with may share class labels.

For instance, rather than:

A B

1 0 # data point is an instance of A 
0 1 # data point is an instance of B

I have a case where the classes may be mixed:

A   B

0.7 0.3 # data point is composed of 70% A and 30% B 
0.3 0.7 # data point is composed of 30% A and 70% B 

i.e. the typical Bernoulli derivation of the binary cross entropy loss is not valid in this case.

Since I do not know much about the topic,I was wondering if there exists literature on the subject of fuzzy class label classification; in particular the loss function.

Furthermore, is my suspicion that Kullback-Leibler Divergence:

is a valid loss in such cases correct? If not, why not?