How do I emulate DiscreteUniformDistribution in Mathematica?

I would use Piecewise.

pdist = ProbabilityDistribution[Piecewise[Map[{1/2, x == #1} &, {0, 1}], 0], {x, 0, 1, 1}]
PDF[pdist, {0.5, 0, 1}]

(*{0, 1/2, 1/2}*)

UPDATE: this problem has been resolved and is working correctly in v11.1.1.0.

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Your syntax:

mine = ProbabilityDistribution[1/2, {x, 0, 1, 1}]; 

... appears to be valid to define a discrete pmf using Mathematica's ProbabilityDistribution function. As such, this appears to be a bug in Mathematica.

The problem you have identified occurs with the PDF function ... for instance:

PDF[mine, 1/3]

1/2

(whereas the answer should of course be 0), but it also occurs with other functions such as the Probability function which also gets it wrong:

Probability[x == 1/3, Distributed[x, mine]]

1/2

Another way to work around this (without using Piecewise) is to explicitly force the discreteness using Boole, as per:

mine2 = ProbabilityDistribution[Boole[x == 0 || x == 1] 1/2 , {x, 0, 1, 1}];

This will then work correctly, ... but it really should not be necessary:

PDF[mine2, 1/3]

0

Probability[x == 1/3, Distributed[x, mine2]]

0

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Another approach is to use the mathStatica add-on to Mathematica, which you can set up exactly as you desired:

f = 1/2;        domain[f] = {x, 0, 1}  &&  {Discrete};

and which fully understands the discrete nature of the pmf:

Prob[x == 1/3, f]

0

etc ...