Consistency of the IID sample mean under a finite first moment

Given: The experiment, finite mean, and IID sequence in the statement.

Identical distribution and IID families: A family is IID when it is identically distributed and the whole family is independent; no moment assumption is implicit.

Partial sums, row sums and sample means: For real random variables $X_1,X_2,\dots$, $S_n={\sum }_{i=1}^n{X}_i$ and ${\bar{X}}_n=S_n\mathrm{/}n$; these finite sums and means are random variables.

Independent random elements have product joint law: A finite independent family of random elements has joint law equal to the finite product of its marginal laws.

Khinchin weak law for integrable IID variables: If $(X_i)$ are IID real random variables and $\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$, then $S_n\mathrm{/}n\to \mathbb{E}{X}_1$ in probability.

Estimator sequences in IID experiments: The size-$n$ sample space is ${\mathcal{X}}^n$ with law $P_{\theta }^{\otimes n}$, and the estimator is a measurable map on that space.

Pointwise consistency in probability: The estimator is consistent at $\theta$ when $P_{\theta }^{\otimes n}\{\mathrm{\mid }{T}_n-\mu (\theta )\mathrm{\mid }>\epsilon \}\to 0$ for every $\epsilon >0$.

Estimands on a statistical model: An estimand is a specified function from the parameter set to its target set.

Expectation of a nonnegative or integrable random variable: For an integrable real random variable, expectation is its finite Lebesgue integral against the probability measure.

Arithmetic and lattice operations preserve measurability whenever they are defined: Finite sums and scalar multiples of measurable real-valued functions are measurable whenever the operations are pointwise defined.

Change of variables for expectation: If a real random variable $X$ has law $P_{\theta }$, $g:\mathbb{R}\to \mathbb{R}$ is Borel measurable, and $g(X)$ is integrable, then ${\mathbb{E}}_{\theta }[g(X)]=\int g(x)P_{\theta }(dx)$.

Statistical experiments: An experiment supplies the observation measurable space and its parameter-indexed family of sampling laws.

Parametric statistical models: The parametric model is the map $\theta \mapsto P_{\theta }$ on that experiment.

The Borel sigma-algebra of a topological space: The Borel sigma-algebra on $\mathbb{R}$ is generated by its open subsets.

The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded: The usual absolute-value distance on $\mathbb{R}$ is a metric.