regular version of conditional distribution

Question

I am trying to understand the concept of regular conditional distribution. I found the definition quit difficult to grasp . This is what i understood . A regular conditional distribution also called a probability kernel is a two variable function (x,A) -> K(x,A) x a scalar and A a set that verifies the following. when we fix x, we have a probability measure, when we fix A we get a measurable function. Can someone confirm this definition ? and give me an example of a non regular conditional distribution .