Suppose $\{X_1,...X_n\}:\Omega \to \mathbb{R}^p$ be i.i.d. random vectors with common probability law/measure $p$, i.e. $Prob(X_i^{-1}(E))=p(E) \forall E \subset \mathbb{R}^p $ Borel measurable.

Consider the random Dirac measures $\delta_{X_i}$, and their average, which is a random probability measure on $\mathbb{R}^p$, defined by $\frac{1}{n}\sum_{i=1}^{n} \delta_{X_i}$. I'd like to know if $\frac{1}{n}\sum_{i=1}^{n} \delta_{X_i}$ weakly converges to the deterministic measure $p$,

i.e. for **every continuous, bounded function** $f: \mathbb{R}^p \to \mathbb{R}$, must we have:

$$\frac{1}{n}\sum_{i=1}^{n} {f(X_i)} \to \int_{\mathbb{R}^p } f(x)dp(x)$$
**as a convergence in probability** of a sequence of random variables $\frac{1}{n}\sum_{i=1}^{n} {f(X_i)}$?

ON a related note, I'd also like to know if the following is true or not:

**If a sequence of random measures converges to a deterministic measure in probability, is it equivalent to have the same convergence almost surely? This question is motivated by the fact a when a sequence of random variables converge to a sonstant in probability, the convergence is a.s.**

P.S. I understand that this question might be elemenrtary to many of you, so some references would be greatly apreciated!

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