This is the usual distinction between a function and it"s values:

This is the usual distinction between a function and it"s values: E is indeed a function (from L1 to R), but E(X) isn"t a function, it"s the value that the expectation operator assigns to the random variable X, and that value is a real number, a constant, not a random variable (not a function on the sample space). So E(X) is very different from g(X), where g is an ordinary function. The latter is a random variable (any function of a random variable is a random variable). So what"s the corresponding fact about conditional expectation?