Let $K(x)$ be a positive-valued function such that
$$\int_{-\infty}^{\infty} K(x)  \ dx = 1$$
and let 
$$K_{\lambda}(x) = \lambda K(\lambda x), \ \ \ (\lambda > 0);$$
that is to say, the family of functions $$\{K_{\lambda}(x)\}, \ \ \  \lambda \uparrow \infty $$
is an approximate identity generated by dilation. 

Suppose $f(x)$ is a measurable function, but we do not know in advance whether or not $f(x) \in L^2$. Assume that for every $\lambda > 0$, the function
$$K_{\lambda}(x)*f(x)$$
is finite valued at every $x$.

Further, we know that there exists a function $g(x) \in L^2$ such that
$$ \lim_{\lambda \uparrow \infty} \int_{-\infty}^{\infty} [K_{\lambda}(x)*f(x) - g(x)]^{2}  \ dx = 0.$$
Does it follow that $$g(x) = f(x) $$
almost everywhere? 

If it makes the question easier to answer, one could add the assumption that $f(x)$ is locally integrable.