Limit of an infinitely fast wiggling curve

The limiting infinitely fast wiggling curve

$${\cal G}= \lim \sin( \omega x),\quad \omega \rightarrow \infty$$

is not a function either: Although the range of its values is $[-1, 1]$, one cannot compute its value ${\cal G}(x)$ for any specific $x$ except $x=0$.

In addition, its average over any interval is zero, and its weak limit (see the discussion of the term in 4.3.2) is zero:

$$\lim_{\omega \rightarrow \infty} \int_{-1}^1 \sin( \omega x) \psi(x) dx =0, \quad \forall \psi(x)\in L_\infty$$ (2)

However, some moments of ${\cal G}$ are nonzero. The second moment is:

$$\lim_{\omega \rightarrow \infty} \int_{-1}^1 \sin^2( \omega x... ...ver 2} + {1 \over 2} \cos(2 \omega x) \right) dx ={1 \over 2}$$

Similarly, higher even moments are non-zero:

$$\begin{eqnarray*} \lim_{\omega \rightarrow \infty} \int_{-1}^1 \sin^3( \omega x... ...rrow \infty} \int_{-1}^1 \sin^4( \omega x) dx & =& {3 \over 8} \end{eqnarray*}$$

and so on.