The standard route to proving the Central Limit Theorem is using Levys continuity theorem to show that the limit of characteristic functions $\phi_k(t)$ converging to some function $\phi(t)$ result in the sequence of random variables associated with those characteristic functions $X_k$ converge to $X$ where $X$ is a random variable and $E[e^{Xti}X] = \phi(t)$.
Is this result possible to prove without measure theory? Or at least a more limited result using restrictions on $X$?
In essence you would want to prove for a sequence of pdf $f_k(x)$ where $\int_{-\infty}^{\infty}f_k(x)e^{xti} = \phi_k(t)$ that if $\phi_k(t) \rightarrow \phi(t)$ that $f(x) =\mathcal{F}^{-1}\{\phi(t)\}$ is a valid pdf and $f_k(x)\rightarrow f(x)$.
For example I have seen proofs of the Fourier Transform inversion theorem without requiring DCT or Lebesgue Integral though even a more straightforward proof using DCT would be nice.
Best Answer
If $X_1,...,X_n$ and $Y_1,...,Y_n$ are all independent meanzero, variance 1, then by replacing only the last in the sum and for some three times differentiable function $f$, by two Taylor expansions centered at $(X_1+...+X_{n-1})/\sqrt n$ you find $$ |E[f(\frac{X_1+...+X_{n-1} + Y_n}{\sqrt n})] - E[f(\frac{X_1+...+X_{n-1} + X_n}{\sqrt n})] | \le \frac{\sup_{t\in R} |f'''(t)|}{n^{3/2}} E[|X_n|^3 + |Z_n|^3] $$ because the zeroth, first, second order terms in the Taylor expansions cancel out by independence.
Now if $X_1.,..,X_n$ are iid with some common meanzero, variance 1 distribution and $Z_1,...,Z_n$ are iid $N(0,1)$ and we apply the above replacement to transform $X_1.,..,X_n$ iteratively to $X_1,...,X_{n-1},Z_n$, then to $X_1,...,X_{n-2},Z_{n-1},Z_n$, then to $X_1,...,X_{n-3},Z_{n-2},Z_{n-1},Z_n$ etc until having replaced all the $X_i$'s with $Z_i$'s, we accumulate $n$ times the error term above so that $$ |E[f(\frac{X_1+...+X_{n-1} + X_n}{\sqrt n}) - E[f(Z)] | \le \frac{\sup_{t\in R} |f'''(t)|}{n^{1/2}} E[|X_1|^3 + |Z_1|^3] $$ where $Z=(Z_1+...+Z_n)/\sqrt n$ has $N(0,1)$ distribution. At this point you have proved that $|E[f(\frac{X_1+...+X_n}{\sqrt n})] - E[f(Z)]|$ converges to 0 as $n\to+\infty$ when the third moment of $X_1$ is bounded for any function $f$ such that $\sup_{t\in R}|f'''(t)|$ is finite.
This proof does not give you the full generality of the CLT, but is quite approachable early in anyone's probability journey, requriring no measure theory background.
Reference: A User's Guide to Measure Theoretic Probability by David Pollard.