I have a hard time understanding some notation about an indicator function: Let the measure space be given by
$ (\mathbb{N},\mathcal{P}(\mathbb{N}),\mu) $ and let $ \mu $ be the normal counting measure. Let $(f_n)_{1 \leq n}$ and $(g_n)_{1 \leq n}$ be function defined in $\mathcal{M}^+$ and by: $$f_n = 1_{\{ n \} } $$
$$ g_n(x) = \left\{ \begin{array}{ll} 1_{ \{ 1 \} } & \quad n \text{ is odd} \\ 1_{ \{ 2 \} } & \quad n \text{ is even} \end{array} \right. $$ I want to find $ \limsup $ of both function, but I can't really understand what the notation of functions means.
To my knowledge $ 1_{A}(x)$ means that it is $1$ if it is in $A$ and $0$ otherwise, but that doesn't apply to this.
Your understanding is correct. The indicator function $1_{\{2\}}$ is defined to be the function such that $f(2) = 1$ and $f(x) = 0$ for any $x \ne 2$.
$$(f_n) = 1_{\{1\}}, 1_{\{2\}}, 1_{\{3\}}, 1_{\{4\}}, \ldots$$
$$(g_n) = 1_{\{1\}}, 1_{\{2\}}, 1_{\{1\}}, 1_{\{2\}}, \ldots$$