Let $E$ be a Banach space, $\mathfrak{M}\_E$ indicate the family of all nonempty bounded subset of $E$, $\mathfrak{N}\_E$ the family of all relatively compact sets, and $Ker \mu=\{X\in \mathfrak{M}\_E$ such that: $\mu(X)=0\}$.



> 
> **"Standard" Definition:**
> A mapping $\mu:\mathfrak{M}\_E\rightarrow \mathbb R^+$ is said to be a [**measure of noncompactness**](https://dml.cz/bitstream/handle/10338.dmlcz/105982/CommentatMathUnivCarol_021-1980-1_10.pdf) (shortly, MNC) in $E$ if it satisfies the following conditions (for any $X,Y,X\_n \in \mathfrak{M}\_E$ and $a\in E$):
> 
> 
> 1. The family $\operatorname{ker} \mu $ is nonempty and $\operatorname{ker} \mu \subset \mathfrak{N}\_E$.
> 2. $X\subseteq Y\Rightarrow \mu (X)\leq \mu (Y)$.
> 3. $\mu (\bar{X})=\mu (X)$.
> 4. $\mu (\operatorname{Conv}(X))=\mu (X)$.
> 5. $\mu (\lambda X+(1-\lambda) Y)\leq \lambda \mu (X)+(1-\lambda) \mu(Y)$ for $\lambda\in [0,1]$.
> 6. If $\{X\_n\}\subset \mathfrak{M}\_E^c$, such that $X\_{n+1}\subset X\_n$ for $n=1,2,...$ and if $\displaystyle\lim\_{n\rightarrow \infty} \mu(X\_n)=0$ then $X\_{\infty}=\bigcap\_{n=1}^{\infty}X\_n\neq \emptyset$.
> 
> 
> 




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Now, an MNC is said to be:


7. With **maximum property** if: $\mu(X \cup Y)=\max \{\mu(X), \mu(Y)\}$.
8. **Nonsingular** if: $\mu(\{a\} \cup X)=\mu(X)$.




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In this question, I want to see if there is an MNC that is With maximum property but **Not** nonsingular.



A MNC with maximum property but not singular
1. First, according to [paper you cited](https://dml.cz/bitstream/handle/10338.dmlcz/105982/CommentatMathUnivCarol_021-1980-1_10.pdf), $\mathfrak{M}\_E$ is **the** [not "a"] family of all nonempty bounded subset**s** of $E$.
2. The examples of MNC's given in the [paper you cited](https://dml.cz/bitstream/handle/10338.dmlcz/105982/CommentatMathUnivCarol_021-1980-1_10.pdf) -- namely, the Kuratowski MNC $\alpha$, the ball measure $\chi$, and the MNC $\mu$ defined in Theorem 1 of that paper -- are all with the maximum property and nonsingular.
3. Perhaps, you rather wanted an MNC with the maximum property but not **non**singular. Such an MNC, say $\nu$, is given e.g. by the formula $\nu(X):=\sup\{|x|\colon x\in X\}$ for $E=\mathbb R$ and all $X\in\mathfrak{M}\_E$.


