In hypothesis tests, we typically have that the null/alternative hypothesis are typically defined as $H_0 : \mu = \mu_0$ and $H_1: \mu \neq \mu_0$ for a two-tailed test or $H_0 : \mu \geq \mu_0$ and $H_1: \mu < \mu_0$ for a one-tailed test, for some population parameter $\mu$. My question is if there is a difference in power for testing instead $H_0 : \mu \neq \mu_0$ and $H_1: \mu = \mu_0$ for a two-tailed test or $H_0 : \mu < \mu_0$ and $H_1: \mu \geq \mu_0$ for a one-tailed test. Essentially, I fail to see why there is always an equality at a point value within the null. I have read that it is because the conditions stated under the null are of more interest to us, or that the distribution induced under the null is easier to compute. Does anyone have any idea what is the main reason an equality sign is usually included in the null?