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I was thinking about the discrete random variable describing the stopping time ($T$: random variable modelling the toss number where he first reaches his target) of a wealthy gambler reaching his target. It is discussed in some detail here: Gambler with infinite bankroll reaching his target and here: Probability that random walk will reach state $k$ for the first time on step $n$. I realized that when the coin is biased against the wealthy gambler, there is a finite chance he will never reach his target. So, if you calculate the summation:

$$sum_{t=0}^infty P(T=t)$$

you will only get $1$ if the coin he is using has a probability, $pgeq frac 1 2$ of heads. Otherwise, the summation above will result in a number less than $1$. Looking at the definition on Wikipedia, no where does it say that the probability mass function should sum to $1$ (emphasis: in the formal definition). However, right outside the scope of the formal definition, it does.

But this would imply that the wealthy gamblers stopping time when $p < frac 1 2$ has no PMF?

Just wanted to get the community's opinion on this.

Also, if we conclude the PMF doesn't have to sum to $1$, is there then any example of a corresponding probability density function that doesn't integrate to $1$? Perhaps the stopping time (defined as reaching a positive boundary) of a continuous time random walk with negative drift?

EDIT: saying that "never reaching the target is included in the possible outcomes" is not satisfying. We are talking about the random variable $T$. This random variable has a certain domain (which includes $infty$). Summing over the domain should give you $1$. Where in its domain should we fit "never reaching the target"? The fundamental problem remains, is $P(T=t)$ the PMF of $T$ or not? If we say it isn't because it doesn't sum to $1$ over all possible values of $T$, then does it mean $T$ doesn't have a PMF?

Let's say that $s=displaystyle sum_{t=0}^infty P(T=t)ne 1$ (which is, as you say, the case when $p<frac12$ and it is possible that the target will never be reached.) In this case when we talk about the probability that $T=t$, (in other words, the probability that the target is reached in $t$ games), we are implicitly assuming that the target is actually reached. Since the probability of that is $s$, the probability that the target is reached in $t$ games, given that the target is reached, is $frac 1 s P(T=t)$, which, of course, sums to 1 as $t$ runs through its domain.

In my comment, I mentioned that saying an arbitrarily large integer is in the domain of a function is not the same as saying that $infty$ is in the domain. Notice that $infty$ is not in the domain of the $P$ in your post. If we define $t=infty$ to mean that the target is never reached, and then define the domain of a mass function $P^prime$ to be $mathbb Z^{ge0}bigcup left{inftyright}$, and $P^prime$ to be:

$$P^prime(T=t)=left{begin{array}{rl}P(T=t),&tinmathbb Z^{ge0}\ 1-s,&t=inftyend{array}right.$$

where $P(T=t)$ is the function in your post and $s$ is as I've defined it above, then this mass function $P^prime$ does sum to 1.

You might find some of my struggles with the probabilities not summing to 1 in a related problem in this thread interesting.

If it is possible not to reach the target, then "never reaching the target" is included in the set of possible outcomes and its probability is one of the values of the PMF. When you add all the values of the PMF, this probability is included and the sum is $1.$

What you are calling the "domain" is actually the co-domain of $T.$
The domain of $T$ is whatever sample space $Omega$ you are using.

What makes $T$ a random variable is that it is a function that takes the elements of the sample space $Omega$ and maps them to outcomes.
The probability of an outcome is the measure of the subset of the sample space whose elements map to that outcome,
and the measure of the entire sample space is $1.$

Consider what it means if the you add up the probabilities of all possible outcomes produced by $T$ and the sum is not $1.$ That implies that there is some part of the sample space (in fact, a part of the sample space with positive measure) that $T$ fails to map to any outcome.
In that case, not only do you not have a PMF that sums to less than $1$;
not only does $T$ not have a PMF;
$T$ is not even a random variable,
because it fails to meet the necessary requirements to be a function
over the sample space.

So if there is a positive chance that the gambler never reaches the goal,
and you want to have a random variable that returns $n$ if and only if the goal is reached at time $n,$ then your random variable $T$ must return something
in the case where the goal is not reached.
You can call that outcome what you like, but you have to include it in the range of $T.$

If you really do not want to do that, an alternative is to define a different random variable that returns the time at which the goal is reached,
conditioned on the event that the goal is reached.
Since you conditioned that variable on the goal being reached,
it only needs to take values that are finite integers
(since those are the possible stopping times).
It is still the case, however, that if you correctly define this conditional random variable, the sum of the probabilities of its outcomes will be $1.$