Integration with respect to a signed measure on $[0,1]$

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Let $x^*$ be a signed measure on $[0,1]$. I just want to make sense of the following steps.

$$\int_{[0,1]}(s\wedge t)x^* (ds)=\int_0^t s x^*(ds)+t\int_t^1 x^*(ds)= \int_0^tx^* ([s,1])ds$$

I just don't understand the last equality. Any help is appreciated!