# Does pivoting a discrete CDF provide a pivot?

In Section 9.2.3 of Casella's Statistical Inference, they base their confidence interval construction for a parameter $$\theta$$ on a real-valued statistic $$T$$ with cdf $$F_T(t| \theta)$$.

1. They first assume that $$T$$ is a continuous random variable. Then by the Probability Integral Transformation, the continuous random variable $$F (T | \theta)$$ is uniformly distributed over $$(0, 1)$$, and therefore is a pivot which can be used to construct a confidence region. 2. They say the situation where $$T$$ is discrete is similar but has a few additional technical details to consider. My question is when $$T$$ has a discrete distribution. In such case, $$F (T | \theta)$$ takes countably many values and is not uniformly distributed over $$(0, 1)$$. Does the distribution of $$F (T | \theta)$$ depend on $$\theta$$, and therefore it is not a pivot? In such case, why can Casella still think it as a pivot, or am I missing something?

Thanks and regards!