As I am reading Wassermann's book All of Statistics, I notice a fine subtlety in the definition of p-values, which I cannot make sense of. Informally, the Wassermann defines the p-value as
[..] the probability (under $H_0$) of observing a value of the test statistic the same as or more extreme than what was actually observed.
Emphasis added. The same more formally (Theorem 10.12):
Suppose that the size $\alpha$ test is of the form
reject $H_0$ if and only if $T(X^n) \ge c_\alpha$.
Then,
$$\text{$p$-value} = \sup_{\theta\in\Theta_0} P_{\theta_0}[T(X^n) \ge T (x^n)]$$
where $x^n$ is the observed value of $X^n$. If $\Theta_0=\{\theta_0\}$ then $$\text{$p$-value} = P_{\theta_0}[T(X^n) \ge T (x^n)]$$
Furthermore, Wassermann defines the p-value of Pearson's $\chi^2$ test (and other tests analogously) as:
$$\text{$p$-value} = P[\chi^2_{k-1} > T].$$
The part I like to ask for clarification is the greater-equal ($\ge$) sign in the first and the greater ($>$) sign in the second definition. Why don't we write $\ge T$, which would match the first quotation of "the same as or more extreme?"
Is this sheer convenience so that we compute the p-value as $1-F(T)$? I notice that R also use the definition with the $>$ sign, e.g., in chisq.test
.