How much worse is weak* convergence vs weak convergence in L1?

Let's say we have a sequence of functions $f_n \in L^1(\Omega)$ for some nice bounded open set $\Omega \subset \mathbb{R}^N$. In general, even along a subsequence, we don't expect a weak limit (in the sense of testing against $L^{\infty}$ functions), so we often work with the weaker notion of weak$^*$ convergence, that is, a Radon measure $\mu$ such that

$$\int_{\Omega} f_n \phi dx \rightarrow \int_{\Omega} \phi d \mu$$

for every $\phi \in C_b (\Omega)$.

My question is this: If we know through some other means that $\mu$ is absolutely continuous with respect to the Lebesgue measure and therefore given by some density $f \in L^1$, so that the above becomes

$$\int_{\Omega} f_n \phi dx \rightarrow \int_{\Omega} f \phi dx$$

then how far is this from being able to say that the $f_n$ are actually weakly converging to $f$? In other words, does the above hold with any $L^{\infty}$ test function, not just $C_b$?

Obviously a simple density argument will not hold because $C_b(\Omega)$ is closed in the $L^{\infty}$ norm, but I am having trouble thinking of a counter-example.

It seems to me that if we could show for any Borel set $E$ that

$$\int_{E} f_n dx \rightarrow \int_{E} f dx$$

then this would suffice- but obviously we do not have this in general for weak$^*$ convergence. Any thoughts?

Abstract counterexample

Let $E \subset \Omega$ be compact, nowhere dense, and have positive Lebesgue measure (like a fat Cantor set). Then $U = \Omega \setminus E$ is open and dense in $\Omega$. I claim that $L^1(U)$ is weak-* dense in $C_b(\Omega)^*$. This follows from Hahn-Banach, because $L^1(U)$ separates the points of $C_b(\Omega)$: suppose $\phi \in C_b(\Omega)$ is such that $\int \phi f = 0$ for all $f \in L^1(U)$. Take $f = \phi 1_U$; then this says $\int_U \phi^2 = 0$, so $\phi = 0$ almost everywhere on $U$. Since $U$ is open this means $\phi = 0$ everywhere on $U$, and since $U$ is dense we have $\phi = 0$ everywhere on $\Omega$.

In particular, this means you can choose a sequence $f_n \in L^1(U)$ converging weak-* to $f = 1_E$. But clearly $f_n$ does not converge weakly to $f$; take $\phi = 1_E$.

(I cheated a little: $C_b(\Omega)^*$ is not weak-* metrizable, so I cannot necessarily choose a sequence $f_n$ converging weak-* to $f$. You can fix this by choosing a compact regular $\Omega_0 \subset \Omega$ containing $E$ in its interior. Then do everything on $C(\Omega_0)^*$ where the bounded sets are weak-* metrizable.)

Hands-on counterexample

Let's work in one dimension. Let $\Omega = [0,1]$ (or any open set containing $[0,1]$, if you prefer). Let $E \subset [0,1)$ be a fat Cantor set which is closed, nowhere dense, and has positive Lebesgue measure $m(E) > 0$. Set $f = 1_E$. Now for each $n$, partition $[0,1)$ into the intervals $I_{n,j} = [(j-1)/n, j/n)$, $j=1,\dots,n$. Set $$f_n = \sum_{j=1}^n \frac{m(I_{n,j} \cap E)}{m(I_{n,j} \cap E^c)}1_{I_{n,j} \cap E^c}$$ so that $f_n$ is supported on $E^c \cap [0,1)$ and has the property that $\int_{I_{n,j}} f_n(x)\,dx = \int_{I_{n,j}} f(x)\,dx$ for each $j$.

Clearly we do not have $f_n \to f$ weakly (take $\phi = 1_E$).

But suppose $\phi$ is continuous on $\Omega$, hence uniformly continuous on $[0,1]$. Fix $\epsilon > 0$ and choose $N$ so large that the oscillation of $\phi$ is at most $\epsilon$ on every interval of length at most $1/N$. Then for any $n \ge N$ and any $I = I_{n,j}$ \begin{align*}\int_I \phi \cdot (f - f_n) &\le \max_I \phi \int_I f - \min_I \phi \int_I f_n \\ &= (\max_I \phi - \min_I \phi) \int_I f \\ &\le \epsilon \int_I f. \end{align*} Summing over $j$ we have $\int \phi \cdot (f_n - f) \le \epsilon \int f = \epsilon m(E)$. A similar argument gives $\int \phi \cdot (f_n -f) \ge -\epsilon m(E)$. So we have $\left|\int \phi \cdot (f_n - f) \right| \le \epsilon m(E)$, and this shows that $\int \phi f_n \to \int \phi f$. So $f_n \to f$ in your weak-* sense.

Let us take $\Omega = (-1,1)$, and $f_n = n \, (\chi_{(0,1/n)} - \chi_{(-1/n,0)}$. It is easy to see that $$\int_\Omega f_n \, \varphi \, \mathrm d x \to 0$$ for all $C_b(\Omega)$, but $$\int_\Omega f_n \, \varphi \, \mathrm d x = 1$$ for $\varphi = \chi_{(0,1)} \in L^\infty(\Omega)$.