Hi everyone, my name is Matthew, I'm an international student studying maths and physics courses in Berkeley. If anyone is in PY105, PY137B or PY112 then cool! Let me know if you want to chat about anything in those subjects :) (or in this course, of course)

Although 'Math' is literally in my name, I believe it probably has measure 0. (Though I haven't been able to prove this…)

Note: In this passage, the terms L-integral, L-measurable refer to Lebesgue integral, Lebesgue measurable etc.

Intuitively, the integral of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is the area between the graph of the function and the x-axis. In order to apply our notions of measure to integration, it makes sense that we first define integrals to output positive reals, just like a measure. Therefore we focus on functions $f: \mathbb{R} \rightarrow [0,\infty)$ which, intuitively at least, should return a positive real after integration.

It is then very natural to proceed with the definition of the Lebesgue integral, which should bring measure and integration together. We first define the Undergraph $\mathcal{U}$ of $f$: $$\mathcal{U}f = \{(x,y) \in \mathbb{R} \times [0,\infty) : 0 \leq y < f(x)\}$$

Then we simply allow the Lebesgue integral to equal: $$\int f = m (\mathcal{U}f)$$

Clearly, for the integral to be defined, $\mathcal{U}f$ must be L-measurable. In which case, $f$ is called a L-measurable function.

As Rasmus Pallisgaard has correctly pointed out, we drop the $dx$ that we usually see in Riemann integration, since we not simply adding boxes of thickness $dx$. As a side note, if $\mathcal{U}f$ is indeed measurable, $m (\mathcal{U}f) = m^* (\mathcal{U}f)$. Thus we are in a sense still just adding boxes, and likely ones that have infinitesimal dimensions.

We can introduce another property, namely that $f$ is a L-integrable function if it both L-measurable, and if its L-integral is finite. Personally, I think the name isn't very appropriate, since it is still possible to find the L-integral of a non-L-integrable function, only the answer is not finite.

A significant practical advantage L-integration has over Riemann integration is that it can work with a much wider scope of functions. For instance, Riemann integration can't handle a function that is discontinuous at an uncountable number of points, however L-integration can deal with this no problem, so long as the undergraph is L-measurable. An L-integrable function can also be unbounded, so long as the L-measure of the undergraph converges to a finite value.

In Fourier theory, we have been focused on the orthonormal basis $\{e^{2\pi inx}\}$ to do most of the work for us. A natural question to ask, is what if we select a different basis in our analysis? The question turns out to open up the theory of the Wavelet Transform, and leads to insights on the famous uncertainty principle.

Define $\psi : \mathbb{R} \rightarrow \mathbb{C}$. Define the family of functions $\{\psi_{jk} : j,k \in \mathbb{Z}\}$ as $\psi_{jk} = 2^{j/2} \psi(2^j x - k)$. We call $\psi$ an orthonormal wavelet if the family $\{\psi_{jk}\}$ is a complete orthnormal basis for $L^2(\mathbb{R})$.

Thus, $f$ can be written: $$f = \sum_{j,k = -\infty}^{+\infty} c_{jk}\psi_{jk}$$

We also define the integral wavelet transform:$$W_{\psi}[f(x)](a,b) = \int_{-\infty}^{+\infty} f(x) \bar{\psi}(\frac{x-b}{a});$$

where $a = 2^{-j}, b = k2^{-j}$ and $c_{jk} = W_{\psi}[f](a,b)$.

A simple example of an orthonormal wavelet is the Haar wavelet $\psi(x) = \psi_{0,0}(x) = \chi_{[0,1/2]} - \chi_{[1/2, 1]}$ where $\chi$ is the indicator function.

We can see that wavelets allow us to localise to particular regions of our function $f$ by translating and dilating $\psi$ with $a$ and $b$. To understand how this localisation relates to the uncertainty principle, consider the orthonormal wavelet $\phi(x) = \chi_{[p,q]}(x)e^{2\pi inx}$ , where $p,q \in \mathbb{R}$ and $p \leq q$. We construct the family of wavelets by: $$\chi_{[p,q]}^{k} = \chi{[p+k(q-p), q+k(q-p)]}$$ for $k \in \mathbb{Z}$. So we use the notation: $$\phi_{k,n} = \chi_{[p,q]}^k e^{2\pi inx}$$ Note that indeed: $\langle\phi_{k,n}, \phi_{l,m} \rangle = \delta_{kl}\delta_{nm}$.

With this wavelet, we can analyse different sections of $f$ independently, and we can build up a picture of