Does the Poisson kernel give a unique harmonic function with given boundary data? In the answer to this question, a helpful Stack user said that the Poisson kernel does not necessarily give a unique harmonic function, given certain boundary data (in particular on the upper half-plane). I had learned that it did, based on the following argument: take the Green's identity $$\int_D u\Delta v - v\Delta u \,\,dV = \int_{\partial D} u \frac{\partial v}{\partial \nu} - v \frac{\partial u}{\partial \nu}d\sigma.$$ Now take $v(x,y)$ to be such that $\Delta_y v = -\delta_x$ and $v(x,y) = 0$ for $y \in \partial D$, and $\forall x \in D$. Suppose $u$ is a harmonic function with (integrable) boundary data $f$. Then the above gives $$u(x) = -\int_{\partial D}f\frac{\partial v}{\partial \nu}.$$ So $u$ is the unique such function, and altering $f$ on a set of measure zero should make no difference.
Is this argument correct?
This arose from trying to show that an analytic function defined on an infinite wedge in $\mathbb{C}$, for which we knew the real part on the boundary (except for two points), is unique up to an additive imaginary constant.
 A: Philosophy
In a course in real analysis / measure theory one learns that sets of measure zero are negligible for integration. And so they are — for integration of functions. But when we integrate an expression involving derivatives of a function, something tricky can happen: if the function is not smooth enough, its derivative may become a distribution that is not represented by a function. Such a distribution may live on just one point (Dirac's delta) and yet contribute to the integral. I already wrote about this difference between straightforward integration and applications of some kind fo Fundamental theorem of Calculus. This is why careful statements of Green's identities (and other forms of the FTC / Stokes' theorem) impose smoothness up to and including the boundary. In specific cases, one may be able to use them for less smooth functions by exhausting the domain by compact subdomains and arguing that the relevant integrals converge; but this convergence is  not automatic.  

Concrete example
The Poisson kernel itself presents an example of how one boundary point can make a difference. The function
$$
u(r,\theta) = \frac{1-r^2}{1-2r\cos\theta +r^2} 
\tag{1}$$
is harmonic in the unit disk $D$ and tends to zero at every boundary point except one. But it is  not  identically zero. It is not  reproduced by the Poisson integral formula from its boundary values, understood as "$0$ a.e."  And in particular,
$$
0=\int_{D} \Delta u \ne \int_{\partial D} \frac{\partial u}{\partial n} =-2\pi
\tag{2}$$
(taking exterior normal). In physical terms, the gradient field $\nabla u$ flows in across the whole boundary, and has no sinks inside of the domain. Where does it go? It squeezes out through the single point $1$.
What do we need to know about a harmonic function $h$ on $D$ to conclude that it's represented by a Poisson integral of an $L^1$ function? A necessary condition is 
$$
\sup_{0<r<1}\int_0^{2\pi} |h(r,\theta)|\,d\theta<\infty
\tag{3}$$
The reason (3) is necessary is that convolution with Poisson kernel is a contraction on $L^1$ space; thus, the $L^1$ norm of $h$ on every concentric circle is at most the $L^1$ norm of its boundary values. But (3) is not sufficient for $h$ to be the Poisson integral of an $L^1$ function, because (1) satisfies it. It turns out that (3) is necessary and sufficient for $h$ to be the Poisson integral of a finite signed measure on $\partial D$. For the function (1), this measure is a point mass at $1$. 
If a  stronger condition holds:  there is $p>1$ such that 
$$
\sup_{0<r<1}\int_0^{2\pi} |h(r,\theta)|^p\,d\theta <\infty
\tag{4}$$
then  $h$ is the Poisson integral of an $L^1$ (in fact $L^p$) function on the boundary. The boundary values can be understood in the sense of nontangential limits, or as a limit in $L^p$ of the restrictions of $h$ to concentric circles. In particular, every bounded harmonic function on $D$  is the Poisson integral of an $L^\infty$ function on the boundary. For example, if we prescribe $h$ to be $1$ on some part of the boundary, and $-2$ on another, and $\limsup_{z\to\zeta}|h(z)|\le 2$ when $\zeta$ belongs to   the remaining  part of boundary (of measure zero), then such a harmonic function is unique.

Remarks
The difference between $p=1$ and $p>1$ has to do with the fact that $L^1$ is not a dual space. For $p>1$, we can use the Banach-Alaoglu theorem  to obtain a weak limit of restrictions of $h$ to circles $|z|=r$; this weak limit gives the notion of boundary values that's suitable for integration. 
Condition (4) can be weakened to uniform integrability of the restrictions of $h$. 
When instead of $D$ we look at the half-plane, we should realize that $\infty$ is a boundary point as well. A Möbius map makes this precise, sending $\infty$ into a boundary point of the disk.
Recommended reading: Potential theory in the complex plane by T. Ransford. 
