The question is :
Given $X = \{a,b\}$, with $M = P(X)$ and $\mu(\{a\}) = 1,\mu(\{b\}) = \infty$, find the dual of $L^1(\mu)$. Is it $L^\infty(\mu)$? (Rudin's Real and Complex Analysis, chapter 6)
For solving this question, I first investigated what $L^1(\mu)$ is. For this, we note that every function $f : X \to \mathbb R$ is measurable, with $\int_X f \ \mathrm d \mu = f(a)\mu(\{a\}) + f(b) \mu({b})$. Under the convention that $0 \cdot \infty =0$, we see that $L^1(\mu)$ consists of all $f$ with $|f(a)| < \infty$ and $f(b) =0$.
In other words, let $f_1$ be the indicator of the set $\{a\}$. Then $L^1(\mu) = \{cf_1 : |c| < \infty\}$.
Now, what is $L^\infty(\mu)$? It consists of functions whose essential supremum is finite. This means that there exists $M$ such that $\mu(\{x : f(x) > M\}) = 0$. By definition of $\mu$, this forces $f$ to be bounded. Hence, $L^\infty(\mu)$ is the set of bounded functions on $X$.
We see that for all $g \in L^\infty(\mu)$, the functional $f \to \int_X fg$ is bounded on $L^1$. So, there is a linear map from $L^\infty(\mu)$ to $(L^1)^*$.
That it is onto is clear, since every linear functional $T$ is determined by its action on $f_1$, therefore if $Tf_1 = d$, then the function $g = df_1$ is such that $Tf_1 = \int f_1 \cdot df_1 = d$. So, these linear functionals coincide on the entire of $L^1$.
However, the map is not injective, since if $g(a) = h(a)$ then $g$ and $h$ produce the same functional on $L^1$, since $\int gf_1 = \int hf_1 = g(a)$. So if $g(b) \neq h(b)$ then they produce the same functional despite being different on a set of non-zero measure. The kernel is equal to every function satisfying $g(a) = 0$. Call it $K$.
Therefore, are we right to conclude that $(L^1)^*$ is isomorphic to $L^\infty / K$?(first isomorphism theorem)?
My only issue is that I've never seen before, this map being non-injective. For the Lebesgue measure, and others I've seen it has been injective(i.e. if $g$ is such that $\int gf = 0$ for all $f \in L^1$ then $g=0$ a.e.), and the image has either been the whole or a part of $L^\infty$.