# Laplace Transform of Dirac Delta, issue with the limits

I am physicist so sorry for being not very formal.

One normally finds the formulas for following Laplace Transforms:

$$\mathcal{L}(1)=\frac{1}{u}$$

$$\mathcal{L}(\delta(t))=1$$,

$$\mathcal{L}(\int\limits_0^t f(s)ds)=\frac{1}{u}F(u)$$,

with $$\mathcal{L}( f(t))=F(u)$$ and $$\delta(t)$$ Dirac Delta function (distribution). From this I would conclude $$\mathcal{L}(\int\limits_0^t \delta(s)ds)=\frac{1}{u}$$.

However, I always use a convention, that $$\int\limits_0^t f(s)\delta(s)ds=\frac{1}{2}f(0)$$ and consequently

$$\mathcal{L}(\int\limits_0^t \delta(s)ds)=\mathcal{L}(\frac{1}{2})=\frac{1}{2u}$$.

Does $$\mathcal{L}(\delta(t))=1$$ assume other convention and it is "fine" always to set $$\mathcal{L}(\delta(t))=\frac{1}{2}$$ in my calculations or could it "spoil" something?