probability of a collection of uniform random variables conditioned on the max value

Question

Just come across this question:

Let $X = (X_1, X_2, . . . , X_n)$ be i.i.d. according to a uniform distribution $U(0, θ)$ for some $θ > 0$. Let $x = (x_1, x_2, . . . , x_n)$ denote a particular realization of $X$. Let $T(x) = \max(x_1, . . . , x_n)$.

Show that $p(X = x | T(x) = t)$ doesn’t depend on $\theta$, where now $t ∈ R$ is some fixed real satisfying $0 < t < θ$.

My Question:

But I think, given $T(x) = t$, since $X$ is continuous random variable, there are still infinitely many $X=x$ that may satisfy $\max(x_1, . . . , x_n)=t$, thus $p(X = x | T(x) = t)$ for any given $x$ should be zero?

Answer 1

My Question:

But I think, given $T(x) = t$, since $X$ is continuous random variable, there are still infinitely many $X=x$ that may satisfy $\max(x_1, . . . , x_n)=t$, thus $p(X = x \mid T(x) = t)$ for any given $x$ should be zero?

Indeed the probability mass will be zero; however to be sensible, the problem should be interpreted to be looking for a probability density function, not a probability mass function.