  The randomized $p$-value, (nonrandomized) mid-$p$-value and abstract
randomized $p$-value have all been recommended for testing a null hypothesis
whenever the test statistic has a discrete distribution. This paper provides a
unifying framework for these approaches and extends it to the multiple testing
setting. In particular, multiplicity adjusted versions of the aforementioned
$p$-values and multiple test functions are developed. It is demonstrated that,
whenever the usual nonrandomized and randomized decisions to reject or retain
the null hypothesis may differ, the (adjusted) abstract randomized $p$-value
and test function should be reported, especially when the number of tests is
large. It is shown that the proposed approach dominates the traditional
randomized and nonrandomized approaches in terms of bias and variability. Tools
for plotting adjusted abstract randomized $p$-values and for computing multiple
test functions are developed. Examples are used to illustrate the method and to
motivate a new type of multiplicity adjusted mid-$p$-value.
