The usual unbiasedness condition of an estimand $g(\theta)$ is this $$E_\theta[\delta(X)]=g(\theta).$$ Here $g(\theta)$ is a real valued function over $\Omega$ whose value is to be estimated. This is already contained in the word estimand. On the other hand $\delta(X)$ is called estimator.
I have no exact hint why this latter is more general, it is just my intuition. This is the core of my question, whether and why the following is more general:
I've considered another condition $$P_\theta[\delta(X)<g(\theta)]=P_\theta[\delta(X)>g(\theta)].$$
What is the relationship between these 2 conditions; is the latter one more general?