Assume that $X_n$ is a sequence of non-negative random variables such that
$$ \Bbb E[X_n] \to c $$
for some constant $c > 0$. From this we want to deduce the $L^1$-convergence
$$ \Bbb E[|X_n-c|] \to 0. $$
Now, finding a bound for this term turned out to be quite the challenge and I'm not really sure how I should prove this, or whether this is even true. Triangle inequality, reverse triangle inequality, none of these things work. I'm at a loss here. Can someone help me out?
Example for measure space, that the claim is not true:
Take $X_n = 1_{[n, n+1)}$ in a $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$ measure space.
$\mathbb{E}(X_n) = 1 \underset{n \to \infty}{\to} 1$
$\mathbb{E}(\lvert X_n - 1 \rvert) = \infty \neq 0$
Example in probability space: Let $(\Omega, \Sigma) = ([0,1], \mathcal{B}([0,1]))$ with Lebesgue measure on it.
Let $X_n = n \cdot 1_{\left[0, \frac{1}{n}\right)}$.
$\mathbb{E}(X_n) = 1 \underset{n \to \infty}{\to} 1$, but
$\mathbb{E}(\lvert X_n - 1 \rvert) \to 2 \neq 0$.