Intuitively, why does the conditional expectation of X given the trivial sigma algebra equals the expected value of X itself?

Question

If we are conditioning X given the trivial sigma algebra then we get the expectation of X, its proof is trivial but intuitively what does this case represent ?

Answer 1

Let $F_0=\text{{$\emptyset, \Omega$}}$ be the trivial sigma algebra for a probability space $(\Omega, F, P)$. Let $X$ be some integrable random variable defined on $(\Omega, F, P)$.

Intuition that can be helpful for conditional expectation is to think of the operation as providing the "best guess" of a random variable given some information (the trivial sigma algebra being no additional information as Henry points out in his comment). This "best guess" is taken to be the expected value given the information available.

So when we look at the conditional expectation $\mathbb{E} [X|F_0]$, as we condition on the trivial sigma algebra, we have no additional information on the value of $X$. As a result, intuitively our "best guess" for $X$ is simply its expected value. In other words: $\mathbb{E} [X|F_0]$=$\mathbb{E} [X]$.

(I am relatively new to posting answers on this forum so happy address any comments / questions for this answer.)